mm-visualization tools for real-time search algorithms

7/30/2019 MM-Visualization Tools for Real-Time Search Algorithms

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Visualization Tools forReal-time Search Algorithms

Yoshi t aka Kuwata1& Paul R. Cohen

Computer Science Technical Report 93-57

A b s t r a c t

Sea rch methods ar e common mechanisms in problem solving. In ma ny AI applica tions,

they a re used with heuristic functions to prune the search space and improve performa nce. In

last thr ee decades, much effort ha s been directed towa rd resear ch on such heuristic functions

a nd sear ch methods by AI community . As it is very ha rd t o build theoretical models for heuristic

functions to predict their behavior, we can often only check their performance experimentally.

In pra ctica l applica tions, it is import an t t o understa nd th e sea rch space an d behavior of

heuristic functions, otherw ise, we cannot figure out w ha t's going on in actua l a pplica tions an d

cann ot cont rol them. These issues ar e crit ical, especia lly in th e field of rea l-tim e problem

solving, in which applicat ions ha ve time constra ints a nd a re required to finish processing within

the given time interva l.

In t his report, visua lizat ion methods ar e intr oduced a s tools to underst a nd the sear ch

spa ces a nd beha vior of heuristic functions. As exam ples of th e usefulness of visualiza tion

methods, A* an d ID A* algorithms a re represented in va rious forms. They can be used to debug

practical a pplica tions tha t use heuristic functions.

1 NTT Da ta Communicat ions Systems, D evelopment Section


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1. Why Visualizations are Important in AI

Looking ba ck on t he hist ory of science, observa tion t ools often play ed a very

import a nt role in scient ific discovery. In t he a ges before th e invention of th e telescope,

for exam ple, w e could not observe pla nets in d eta il. We didn't even know of the

existence of pla nets. Telescopes ma de it possible to mea sure a ccura te positions of th e

planets, a nd t ha t ma de it possible to find Kepler's law , New ton's law s a nd so forth . The

sa me story is t old in other a rea s of science. The microscope, X-ra y, spectr um a na lyzer

a nd other observa tion tools all cont ribut ed to th e progress of science. In oth er words,

th e history of science is a lso the hist ory of the engineering of observa tion t ools.

The story is t he sa me in comput er science. Today , computers a re used to

simula te a va riety of problems in comput er science; for example, computer simula tions

of th e execut ion queue of an operat ing syst em. Queuing th eory provides some guida nce

but t he optima l queue length is often determined experimenta lly with computer

simula tions. In t his exa mple, comput ers ar e used as t ools to stu dy th eories in comput er

science a nd t hey a re also the ta rget of st udy in comput er science. When w e develop

th eories of comput er science, we a lso need t o develop observa tion t ools for t hem. Aga in,

w e ca nnot develop theories w ith out observing phenomena .

In comput er science, especially in a rt ificia l int elligence, th e development of

observa tion t ools seems not t o ha ve been very focused. This pa per describes a collection

of visualization tools for search algorithms, because search algorithms are very common.

To apply sea rch mecha nisms for problem solving, the problem spa ce must be defined a s

a set of sta tes, including an initial sta te a nd goal sta tes, and a set of operators to move

from one sta te to th e oth er sta tes. A solution corresponds t o a s eries of opera tions

moving from the initial sta te to a goal sta te.

We can a na lyze sear ch a lgorith ms theoretically or empirically. Theoretica l

a na lyses ar e sometimes bett er tha n experiment a l ones, because w e don't need to collect

da ta , and beca use the results a re often more genera l . However, we usua lly need to rely

on heuristic functions t o solve big an d complex problems in realist ic time. H euristic

functions ma ke a na lysis ha rder as th ey often don't ha ve theoretical underpinning. In

th e ca se of chess, for exam ple, th ousan ds of suggestions exist in t he books, but t hey

don't ha ve theoretica l explana tions. Inst ead, th ey ca me from huma ns' experiences and

inspira tions. In such cases, we need to rely on experimenta l an a lysis of the heuristics.

B y a pplying visualizat ion t ools to search a lgorithms, w e can observe phenomena

caused by heurist ic functions. These tools ar e also useful for debugging, verifica tion a ndvalidation.

Sea rch spaces a re described in section 2 of th is paper. Convent iona l techniq ues

such as search t ree representa tions an d depth-number of node representa tions, and

their adva nta ges a nd disadva nta ges a s visualizat ions are discussed in section 3. In

section 4, visua lizat ions of heuristic function are described w ith exa mples from th e


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Visual i zat i on Tool s for Real -t ime Sear ch Algor i thms Kuwata & Cohen

2

Eight P uzzle. Time series and frequency an a lysis of search a lgorithms a re discussed

w ith exa mples in section 5.


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2. Objects to Visualize in Search

2.1 Search Space

The most importa nt thing in problem solving by sear ch is to understa nd sea rch

spa ces th emselves. They va ry from problem to problem. For example, backgam mon ha s10

20sta tes in its search space a nd chess has a much bigger stat e spa ce. The structure

of sear ch spaces is a lso completely different in va rious problems. In chess, lar ge

numbers of actions a re possible in th e f irst sta ge of th e gam e but a ctions a re l imited in

the endga me. On the other ha nd, in shogi, which is a t wo-player gam e similar t o chess

tha t a llows reuse of dead pieces, the movements in th e last sta ge aren't l imited a s in

chess.2 Therefore, the sea rch spa ces for the tw o games w ould be different, even though

th ese ga mes look simila r.

The simplest w a y t o represent a search space is to count t he number of sta tes in

it . For exa mple, we can compar e tw o problems A a nd B , which ha ve ten million st a tes

a nd one million st a tes, respectively. We expect to t a ke 10 times longer to solve problemA th a n to solve problem B. This is because w e assum ed problem A a nd B a re equa lly

difficult; i .e., they ha ve the sam e sea rch space structu re. B ut it is possible to define a

problem wh ich has a bigger search spa ce but is simpler to solve. Thus simple search

sta te count a lone is not sufficient t o cha ra cterize a search problem.

Which problem is bigger ?

Go a l

Go a lA B

Figure 2.1.1 Two Problems The num ber of possible stat es in problem A is much sma ller

tha n in problem B . It t akes more tha n 100 steps to reach the goal in problem A, but only 3

steps in problem B on a verage.

2.2 Shape of search space (Branching Factor and Depth)Assuming w e ca n represent problem spa ces a s sear ch trees, the a verage

bran ching factor an d depth of sea rch tree a re useful cha ra cteriza tions. The avera ge

bran ching fa ctor is the a verage number of successor sta tes following ea ch sta te. For

2 Although possible actions in the endgam e are fewer th an in the first sta ge, shogi ha s more possible

endgame a ct ions tha n chess.


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example, in the E ight P uzzle, the number of successor nodes3 is either one, tw o, th ree or

four. When w e solve the puzzle allowing cyclic a ctions4, the a verage bran ching factor is

a bout 2.6. On the other ha nd, it is 2.3 w ith out cyclic a ctions. These num bers were

ta ken from a ctual execution tr a ces of searches tha t ha ve different successor functions.

It is often very difficult to ca lcula te the theoretical bra nching factor. Thus experiment a l

values for bran ching fa ctor5

a re often used. In general , the bran ching factor dependsupon th e successor functions 6 in the search method.

The depth of a sea rch tr ee represents t he length of a sequence of opera tions

required to reach the termin a l nodes. When w e a ssume a sea rch tree is completely

balanced a nd ha s branching factor band depth d, th e number of nodes at level i is given

by bi , therefore th e tota l number of nodes in the tree n is given by th e follow ingequation:

n= 1+ b+ b2 + b3 + L + bd

= bii=0

d

(2.2.1)

If we a ssume a t ree ha s different bra nching fa ctor bi a t level i, equa t ion (2.2.1)7

w ould be the follow ing:

n= 1+ b1+ b1b2 + b1b2 b3 + L + b1b2 b3Lbd

= 1+ bjj=1

i

i=1

d

(2.2.2)

E qua tions (2.2.1) a nd (2.2.2) can be applied to a fully ba lan ced t ree only w hen

theoretical bra nching factors a nd sear ch depth a re known . However, i t is possible todetermine th e effecti ve br anching factora t level i( bi) experimentally by actuallysear ching. When we count t he number of nodes at level i a s ni and at level i +1a s ni + 1,the a verage effective branching fa ctor a t level iis defined by th e follow ing equa tion:

bi =ni + 1

ni(2.2.3)

The chan ge in effective bra nching fa ctor shows th e cha nge of th e eff ecti ve sear ch

spacewh ich is defined as a result of the search spa ce a nd of the h euristic function used

to prune nodes. For exa mple, w e w ould observe lar ge ba t sha llow levels but small bin

3 Successor nodes in a search t ree represent possible successive stat es.4 Cyclic actions are defined to generat e the same stat es that a ppeared previously. For exam ple, moving one

piece back an d forth ma ny t imes corresponds to a cyclic action. We know such actions would not help in

solving t he puzzle.5 We refer to experimenta l bran ching factor as th e "effective bran ching factor" in t he rest of this paper.6 Successor functions a re functions tha t r eturn successor nodes.7 This equat ion still assum es the tree is bala nced but has different bra nching factor at ea ch level.


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deep levels in th e sea rch of gam es like chess an d shogi. In such ga mes, there are a huge

number of possible actions in the beginning st a ges but fewer a t t he end. Thus t he

expected branching fa ctor in search a t each level will vary.

Note also tha t equat ion (2.2.3) gives only the a verage bra nching factor a t a

cert a in level a nd does not tell us w heth er the search tree is ba lan ced. The height ofsearch trees is not n ecessarily the sa me at each bran ch in actua l sea rch, as one bran ch

can reach a dead end before a nother. In such ca ses, the number of terminal nodes a t

level i will a f fect the ca lculat ion of bra nching fa ctor a t t ha t level. B eca use terminal

nodes ha ve no successor nodes, their bran ching factor is 0, w hich decrea ses the avera ge

bran ching factor. When we ar e interested only in the nodes tha t ha ve a t least one

successor node, we should exclude termin a l nodes from our calcula tions. One exa mple

of th e effect of termina l nodes on the a vera ge bran ching factor is shown a s figure 2.1.2.

depth1

depth2

depth3

Figure 2.1.2 An example of an Unbalanced Tree In t his tree, there is one node ateach depth w hich has 8 successor nodes while all others ar e termina l nodes (bra nching

factor = 0). The avera ge branching factor at depth 2 and depth 3 is 1. Nodes wh ich ha ve 8

successor nodes contr ibute to increase the a verage bra nching factor.

This effect is a lso observed w hen w e use a h euristic function to prune bra nches

from a search tr ee. The pruned nodes can be regard ed as a termina l node, therefore, the

effective branching fa ctors a t each depth becomes sma ller t ha n t he values given by t he

successor function. This effect is a lso observed w hen w e ar e forced to stop sea rch a t a

cert a in level. In t his ca se, th e observed bran ching factor at th e low est level would be

zero.

2.3 Pruning and Heuristics

When it is known tha t some pa rts of the search space are not worth searching,

th e cost of sea rch ca n be sa ved by excluding t hose a rea s from th e solution space. For

exa mple, in th e search a lgorithms used to solve the Eight P uzzle, we don't need t o

consider th e previous st a te a s a successor st a te, beca use it w ould cause mea ningless


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itera tive a ctions. We know su ch actions w ould not help us solve the puzzle, th erefore,

we can prune them.

It is common a lso to use heurist ics to choose th e best su ccessor st a te ea rlier tha n

less worth y successors. If th e problem is to find one accepta ble solut ion, a nd not t o

sear ch all of th e sta te spa ce, choosing th e best a ction first w ill help us find t he solutionfa ster. H euristics a re used for tw o purposes:

1. to choose th e best successor sta te

2. to omit subt rees where t here is no need t o sea rch (prunin g)

Visua lizat ions must be capa ble of representing both of th ese purposes. For

example, we ca n use bra nching fact or a na lysis directly t o know th e effect of prun ing. As

the result of pruning, the a verage bran ching fa ctor becomes sma ller t ha n t he expected

value. On the other ha nd, we can indirectly observe th e quality of the pa th t he heuristic

function chooses from th e sequence of heuristic values the function return s. In t he ca se

of h functions in A* sea rch, for exam ple, if the fun ction is chooses a good pa th to th e

goa l, th e h va lue is expected to decrease.

A good visualization would represent these two purposes directly and separately.

2.4 Characteristics of search algorithms

As th e goa l of visualizat ions for sea rch algorithms is to understa nd t he

a lgorithms a nd t o predict their behaviors, visualization tools must h a ve the abili ty to

represent how sea rch algorithm s a re working.

For example, depth-first search8 a nd breadth-first search9 behave

differently. The former sea rches dow n to the deepest pla ce first w hile th e la tt er fully

explores th e sha llow est level of th e search t ree first . We ca n decompose th e

char a cteristics of oth er search a lgorith ms int o depth-first component s (deeper first) and

bread th -first components (broa der first ). Best-first search10 is a general s tra tegy to

sear ch the best possible successor node first . If th e eva lua tion function of a best-first

sear ch relies heavily on the cost of the pa th t o rea ch the node, the search w ill be simila r

to brea dth -first search. B y contr a st, i f the evalua tion function emphasizes the

rea cha bility of goa ls, the sea rch would be similar t o depth-first sea rch. The

char a cteristics of best-first sea rch depend on the evalua tion function. In genera l, in

best-first search, we need to balance various cost factors in the evaluation function in a

ta sk-specific ma nner.

Eva luat ion functions typica lly must be ca refully tun ed by experiment.

Decomposing search beha vior int o depth-first a nd brea dt h-first component s is one wa y

8 P seudo-code of depth -first sea rch is show n in Appendix A.1.9 Pseudo-code of breadth-first search is shown in Appendix A.1.10 P seudo-code of best-first sea rch is show n in Appendix A.1.


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to chara cterize a nd understa nd sea rch a lgorithms. In this paper, var ious visua liza tion

techniqu es ar e shown t o accomplish t his decomposition.

There a re intera ctions betw een search algorithm s and sea rch spa ces. Eva luat ion

functions t ha t a re ca refully tuned to one part icular problem w ill not a lwa ys work well

for slightly different problems. For exa mple, th e right -ha nd meth od11

, or left ha ndmeth od, a re ea sy w a ys to solve simple ma ze problems. They a re basica lly a depth-first

sear ch. Figur e 2.1.3 is a simple example of tw o ma ze problems, wh ich illustra tes a n

intera ction effect betw een sear ch algorithm s an d search spa ces. In cas e (A), the left -

ha nd meth od w orks better tha n th e right-ha nd method; in ca se (B ) the right -ha nd

method works better.

start

goal

start

goal

(A) (B)

Figure 2.1.3 Two Maze Problems One simple example of the int eraction of algorithman d search space. (A) is easily solved by left-ha nd meth od. On the other ha nd, in ma ze

(B ), w hich is th e mirror ima ge of (A), left-ha nd m ethod is not a good solution.

Actual observa tions depend both on the char a cteristics of search a nd t he search

spa ces. A good visualiza tion would represent th em independent ly and help determ inethe int eraction effect betw een t hem.

11 One can solve simple mazes, which ha ve no islands, by keeping one hand to the wa ll and w alking a round

the maze. "R ight-ha nd" m ethod refers to solving the problem by using the right-ha nd and t urning ar ound

the ma ze clockwise, and vice versa for "left-ha nd" method.


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3. Conventional Visualization tools

3.1 Search Trees, Graphs

It is very common to represent sea rch spaces as t rees 12. In this representa tion,

nodes in th e tree correspond to sta tes in th e sea rch. Sea rch goes from the root node ofth e tree down to t he leaf nodes nodes w hich ha ve no furt her successor n odes or w hich

a re desired sta tes. The pat h from the root t o the leaf nodes traces the steps from the

sta rt st a te to a goal sta te in the solution space. The following f igure shows a tree

representa tion of a search space.

Figure 3.1.1 Example of Graph Representation of Search Tree Fully balanced treewit h bra nching factor b= 2 and height of tree = 4. There are 31 tota l nodes in this tree,

including root node a nd lea f nodes.

In t his representa t ion, when w e draw arcs as the same length an d descendantnodes as t he same dista nce, we ca n easily see the search depth an d search widt h a s the

height a nd t he widt h of the sear ch tree, respectively.

I t is a lso easy t o show chara cteristics of search a lgorithms such a s dif ferences in

sear ch order. For exa mple, depth-first sea rch will pick up one pat h a nd cont inue

searching down to the lea f node, which is a sta te the sear ch algorithm wa s looking for,

or w hich ha s no successors. In t he former case, the sea rch algorithm ha s the option t o

keep searching un til it finds bett er solutions. To sear ch furth er in the tr ee a fter

reaching th e lea f node, the sea rch a lgorithm needs to look up other bra nches in the

pat h. The simplest meth od is to sear ch lowest a ltern a tive bran ches. This operat ion is

known a s backtracking. Depth -first sear ch w ould cont inue ba cktra cking unt il it finds asolut ion or searches a ll of the tr ee. The follow ing figure shows t he cha ra cteristics of

depth-first search using tree representations.

12 In genera l, a stat e space would be represented by genera l directed gra phs. However, we can expan d

these graph representa tions by duplicat e subtrees in a t ree representa tion.


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1

2

3

4 5

6

7

8 9

10

Figure 3.1.2 Depth-first Search in a Search Tree The order of sea rch is la beled oneach of the bra nches. In th is exam ple, the search a lgorithm checks the leftmost bra nches

first , then goes to the right bra nches.

On th e other ha nd, a breadt h-first search a lgorithm searches the sha llow level of

th e tree first , then increa ses the level of search one by one. Figur e 3.1.3 show s th e

beha vior of breadt h-first sea rch using t he sa me representa tion a s depth-first sear ch.

1

2 3

4 5 6

7 8 9 10

Figure 3.1.3 Breadth-first Search in a Search Tree The order of sea rch is la beled oneach of the bra nches. B read th-first search checks nodes in the sha llowest level first , then

searches progressively deeper levels.

These representa tions are very intuitive and ea sy to understa nd, but t hey a re

useful main ly for sma ll sea rch spaces. When th e tree becomes lar ge, w e ca nnot

represent a ll search sta tes as n odes in th e sea rch tree. P a rt of this diff iculty depends on

huma n perceptua l ca pabili t ies. I t is sa id to be ha rd for huma ns to hand le more tha n 10

to 20 objects a t once. The tr ee representa tion of th e search spa ce for a pra ctical problem

could be more th a n ten million sta tes. Even if we could recognize pat tern s in such a

large tree representa tion, we could not understa nd t hem in deta il . To determine the

deta ils of th ese pa tt erns, we need to look closely a t individua l nodes a nd pa ths inside

the t ree.

In t he lar ge tree in figure 3.1.4 (showing a bout tw o hun dred nodes) w e ca n

recognize tha t t he top leftmost bra nch ha s ma ny more nodes tha n other tw o bra nches.

However, we ca nnot determine the a verage bran ching fa ctor of ea ch branch from this

representation.


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Figure 3.1.4 Large Tree This figure is made by ra ndomly pruning subtr ees from a fullybala nced tree with b= 6 an d h= 7. The resulting tree includes about 200 nodes.

This discussion is a lso va lid for va ria tions of tree representa tions such a s 'AND -

OR trees' a nd ' la beled trees. ' They a re very int uitive and ea sy to understa nd for sma ll

search spaces. Thus th ey help us understa nd th e ba sis of sea rch spa ces an d a lgorithms,

but a re less useful for a na lyzing large sca le sea rch spaces in deta il .

3.2 Depth-number of node counts

The size of a sea rch spa ce is commonly represented a s t he num ber of nodes a t

each level.

Theoretical Depth-Node CountsFigur e 3.2.1 plots n ode count s for a sear ch tree. The number of nodes at each

level is given by bi , where iis the depth of the search tr ee a nd bis the a veragebra nching fact or. Depth -node count s for th ree va lues of ba re shown in this f igure.

4 5 6 7 8 9 10

2000

4000

6000

8000

b=2.3

b=2.5b=2.8

Depth

#Node

Figure 3.2.1 Node Count Representation of Search Trees This figu re is ba sed onequa tion (2.2.1). The number of nodes at successive depths increas es exponentia lly.


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Empirical Depth-node countsI t is also useful to apply the node count visualizat ion t o empirica l dat a . The

following f igure shows a ctua l dat a from E ight P uzzle search using three different search

meth ods. One is pure depth-first sea rch wit h a d epth-bound, which corresponds to one

iterat ion of IDA* search with a consta nt h function. The oth er tw o a re IDA* sear ch13:

one uses a Ma nha tt a n dista nce function forh a nd t he other uses a f inal position count

function for h .14

0

5000

10000

15000

20000

25000

30000

35000

40000

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

Dept

n

umberofnodes

pure depth-first search

IDA* withfinal position count

IDA* withManhattan-distance

Figure 3.2.2 Empirical node counts from Eight Puzzle This figure shows t he number

of nodes expanded a t ea ch level of the search tree for three a lgorithms. As pure depth-firstsearch doesn't prune nodes at all, th e curve from pure depth-first sear ch shows the t otal

number of nodes at ea ch level. B y compar ing with t ha t curve, we can see how ma ny nodes

ar e pruned by the other tw o algorithms. If the number of nodes at one level is the sam e as

pure depth-first sear ch, then there is no pruning a t th at level.

The curve from pure depth -first sear ch uses no prun ing, t herefore it represents

a ll of the nodes a t ea ch depth. This curve grows exponentia lly as w e have shown in t he

theoretical depth-node counts graph (figure 3.2.1).

Now w e ca n check the effect of pruning by comparing pruned dat a w ith t he pure

depth-first sea rch curve. The oth er tw o curves a re almost ident ical to the no-pruning

curve until depth = 9. This mea ns the tw o a lgorithms expanded almost the sam enum ber of nodes unt il depth = 9. The curve from IDA* with th e final position count

function is different from no-prunin g curve only from depth = 14, but I DA* with

Man ha tt a n dista nce ha s a lower growth ra tio tha n the no-pruning curve. This shows

tha t t he former can prune nodes only a t t he deepest levels, but t he lat ter can prune

13 P seudo-code of ID A* sea rch is sh own in Appendix A.3.14 These heuristic functions are described in Appendix A4 in detail.


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nodes a t sha llow levels a lso. We cann ot know why t hese functi ons are dif fer ent , given

th i s r epr esent ati on, but we can k now how they are di ffer ent . We need to introduce

visualization of heuristics to know why these heuristic functions behave differently.

From equa tion (2.2.3) the a verage bra nching factor a t each level for t his da ta can

be ca lcula ted. The result is show n in figure 3.2.3.

0.5

1.0

1.5

2.0

2.5

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

Dept

pure depth-first search

IDA* with final position count

IDA* with Manhattan-distance

averag

ebranchingfactor

Figure 3.2.3 Empirical branching factor at each level of the Eight Puzzle Thisfigure shows the empirically derived avera ge branching fa ctor for each algorithm.

I f th e tree is fully bala nced a nd t here is no pruning during sear ch, the average

bra nching fa ctor is expected to be consta nt a t every level a nd t he plot is expected t o be a

flat l ine. In sear ch without pruning, which is shown a s pure depth-first search in the

figure, the average branching factor has some structure: sma ll bran ching factors are

followed by lar ge branching factors and vice versa . In t he Eight P uzzle, the full search

tr ee is likely to ha ve a sm a ll number of moves follow ed by a la rge number of moves, a nd

vice versa . For exa mple, th e cent er position ha s thr ee possible moves th a t lead t o sta tes

wh ich ha ve only tw o legal m oves.

Time Trace of Node Count

It is also useful to observe the behavior of va rious search meth ods in a nodecount gra ph by plotting t he history of sea rched nodes. Figur e 3.2.4 represents t he

expected hist ory of node count s in depth -first sea rch. In t his gra ph, the number of

nodes at ea ch level increases a s time passes. Thus la ter curves are a bove ea rlier curves,

a s more nodes ar e sear ched a t ea ch depth . The curve grows vertically from bottom t o

top.


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Numbero

fNodes

time

Depth

Figure 3.2.4 Time Trace Representation in Node Count (general depth-first search)

This is th e expected gra ph on time t ra ce of node count in depth-first sea rch. As time pa sses,the num ber of nodes visited by depth-first sear ch increases at each level.

An a ctua l depth -first sea rch for th e Eight P uzzle, show n in figure 3.2.5, is pretty

much as expected. The number of nodes at ea ch depth increases as tim e pa sses. If the

search tr ee is not ba lanced a t a ll , the number of nodes shown in this f igure wouldn't

increase proportional t o time spent for sea rch. We ca n conclude, th en, tha t t he search

tree in Eight P uzzle is balan ced.

2 4 6 8 10

Depth

0

50

100

150

200

250

300

350

NumberofNode

s

Figure 3.2.5 Time Trace Representation in Node Count (Depth-first Search on theEight Puzzle) This gra ph wa s ma de from one insta nce of a t ime tra ce of depth-first sear ch onthe Eigh t P uzzle. Ea ch curve represents th e number of nodes at each depth in every 80 nodes

expanded; i.e. , the lowest curve is t he plot of the first 80 nodes searched, t he second lowest

curve is from the first 160 nodes, a nd so on. The envelopes of these curves grow exponentia lly

as the depth increases.


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14

On the other hand, the graph of breadth-first search will be completely different

from depth -first sea rch in a n ode count r epresent a tion wit h history. The follow ing

figure is an expected execution history of depth -first sea rch. In t his represent a tion, the

breadth-first search progresses horizontally from left to right. This is because th e

sear ch moves from sha llow levels to deeper levels progressively. If t he sear ch space is

th e sam e as for depth -first sea rch, the envelope of th e curve is also the sa meexponentia l curve as t ha t of depth-first sear ch.

Num

berofNodes

time

Depth

Figure 3.2.6 Time Trace Representation in Node Count (general breadth-first search)As time passes, th e frontier of the sea rch progresses from left to right horizonta lly. The

envelope of the curve reflects the sear ch space and is the sa me w ith d epth-first curve.

The result of a n a ctual brea dth -first search on the E ight P uzzle is shown in

figur e 3.2.7.

2 4 6 8 10

Depth

0

50

100

150

200

250

300

350

NumberofNodes

Figure 3.2.7 Time Trace Representation in Node Count (Breadth-first Searchon Eight Puzzle) This graph w as ma de from a t ime tra ce of breadt h-first search onthe Eight P uzzle. Ea ch curve represents the number of nodes at ea ch depth.


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Other search a lgorithms m ix breadt h-first search a nd depth-first search. The

follow ing figure shows best -first sea rch a pplied t o the sa me sear ch tree a s figures 3.2.4

a nd 3.2.5. If th e eva lua tion function used in best-first sea rch relies on depth, th e sea rch

is like depth -first sea rch a nd t he envelope grow s vertically. If th e function relies on

breadth , the search becomes like breadt h-first search a nd curve grows horizonta lly . In

other w ords, th e ver ti cal growt h of th e cur ve r epr esent s depth -fi r st component s in t hesear ch and th e hori zont al growt h of t he cur ve r epr esent s breadt h-fi r st component s.

Depth

Nu

mberofNodes

time

Figure 3.2.8 Time Trace Representation in Node Count (general best-first search)G eneral best-first search ha s a depth-first component (vertical growth ) an d a br eadt h-first

component (horizonta l gr owth).

The result from ID A* sea rch with Ma nha tt a n dista nce is shown in f igure 3.2.9.

Ea ch curve corresponds to a n itera tion of ID A* search; i.e., the leftmost curve, wh ich

sta rts a t depth 0 and reaches depth 3, is from the f irst i t erat ion. The second leftmostcurve is from th e second itera tion, wh ich expan ded nodes to level 4, an d so fort h. The

eighth (last ) i terat ion is shown a s the t op curve. In t his case, a solution is found a t t he

eighth it era tion a t level 21. The search horizon progresses horizonta lly in the lat er

iterat ion of IDA* sea rch. This fact shows t ha t t he heuristic function prunes ma ny

bran ches in deeper pa rts of the search tr ee.

For comparison, a non-pruning a lgorithm is a lso shown in this f igure a s a n

exponentia l curve. The sear ch is very mu ch like depth-first sea rch (i.e., it grows r a pidly

in the horizontal direction) but expands fewer nodes than pure depth-first search.

Especially in the last tw o itera tions, the sea rch progresses a t more tha n ten levels at

once. This means tha t t he heuristic function can help choose the right bra nch an d a voidsearching bad pat hs .


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0 5 10 15 20

Depth

0

10

20

30

40

50

60

7

Number

ofNodes

No pruning

8th iteration

1st iteration

Figure 3.2.9 Time Trace Representation in Node Count (IDA* Search with Manhattandistance) One insta nce of a t ime tra ce of node count using ID A* sear ch on the E ight P uzzle.

Figur e 3.2.10 is an other exam ple from the puzzle wit h a different h function. In

th is ca se, the left-ha nd side of the envelope of the t ime tra ce follow s exactly t he

exponentia l no-pruning curve, w hich m eans the a lgorithm expa nded a ll nodes a t each

level wit hout pruning. From compar ison w ith t he previous f igures, this gra ph a lso

shows this search includes more breadth-first components than the previous search

a lgorithm, a s t he envelope of the curve grows fast er vertica lly tha n h orizont a lly

(compa re with figur e 3.2.9). This mean s tha t t he h function is not helping very much by

pruning nodes, an d a s a result , sea rch ca nnot go deeper int o the search tree until i t

expands ma ny more nodes at sha llower depths.

5 10 15 20

Depth

0

250

500

750

1000

1250

1500

75

NumberofNodes

No pruning13rd iteration

1st iteration

Figure 3.2.10 Time Trace Representation in Node Count (IDA* Search with final

position count) Another insta nce of a t ime tra ce in node count usin g ID A* search on theEight P uzzle. Note tha t the scale of the Y-axis is much bigger than in th e previous tw o figures.


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4. Visualization of Heuristics

4.1 Depth-Heuristic Function Representations

When heurist ic functions a re used for sea rch, visualiza tions of th e sea rch give us

ma ny useful ideas a bout how to evalua te them. In ma ny cases, we don't ha veth eoretica l expecta tions for heuristic functions. We need to determine how w ell th ey

perform empirically, for w hich we can r ely on visua lizat ions of them.

The depth -heur i sti c fun cti on repr esent ati onis shown in th e rest of this

subsection. Figure 4.1.1 wa s ma de from a history of search in solving the E ight P uzzle

problem. This figure is from one par ticula r sea rch. The X-a xis represents h-value

(estima ted h-va lue) given the current s ta te, th e Y-a xis shows th e depth of th e curr ent

sta te from th e initia l sta te, and t he gray level at each point represents t he number of

sta tes a t t ha t level. The positions colored in black signify tha t no such st a te exists in

the search space. White sta nds for ma ny nodes15. In this f igure, th e h function was

used only t o evalua te each st a te; i t w a sn't used during sear ch to prune successor nodes.Thus t his figure represent s th e structu re of the ent ire sea rch spa ce from the view of th e

h function. For example, the h-va lue of the initia l condition wa s 16. Then tw o nodes at

h-va lues 15 an d 17 a nd level 2 a re expand ed, next h-va lues 14, 16, an d 18 at level 3,

a nd so forth 16. From th is figure, we can observe th a t t he number of successor nodes

increases as t he sea rch progresses. In pa rticular, a h uge number of nodes ha ve h-

values a round 17 (wh ite place at h= 17, level= 12). The nodes distribut e almost

symmetrically with th e center at h = 16.

In t he actual IDA* search tha t uses this h function, nodes are sea rched in order

of sma ller (h+ g)-value17. Figure 4.1.2 is a visua lizat ion of sear ch order using th e sa me

h function as in 4.1.1. Whit e diagona l arr ows from the bott om right to top left show

groups of nodes w hich ha ve the sa me ( h+ g)-va lue. In th e first itera tion of ID A*search, the nodes on t he leftmost a rrow a re sea rched, because they ha ve the same

( h+ g)-va lue as th e init ia l threshold (16). In t he second itera tion, the th reshold valueis set t o 18 and nodes on the second leftmost a rrow a re also sea rched, a long w ith t he

nodes on the leftmost a rrow. In ea ch iterat ion of IDA* search, the search horizon

progresses one arrow toward the left, as more nodes are searched at deeper levels.

15 It is possible to use a 3D gr aph t ha t shows t he number of nodes on the Z-axis (instead of a 2D gra y scale

gra ph). The gray sca le representa tion is chosen for its intuitiveness of representat ion. In th is figure, gra y

level is determined by th e number of nodes at ea ch stat e and plotted on a logarit hmic scale. (While the

most frequent st at e has more tha n 20,000 nodes, it is pra ctically impossible for human s to distinguish more

tha n 10 gra y levels simulta neously.)16 In t his h-ha t-function, the h-ha t-value of the next sta te is one plus or one minus t he current h -ha t-value.

As the result of this property, the depth h -ha t-function representa tion ha s a checkerboard a ppear an ce.17 In t his example, the depth from th e root node wa s used as t he g-value for ID A* search.


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0 5 10 15 20

0

2

4

6

8

10

12

H-hat-value

Depth

Figure 4.1.1 Example of Depth-Heuristic Function Visualization This figure is ma defrom one instan ce of search on the E ight P uzzle by using the Manha t ta n dis tance with t he

blank space as h funct ion. The X- a nd Y-a xes represent h -value and depth respectively. Ea chsqua re in this plane represents the sta te corresponding to the values on the X-axis a nd th e Y-

axis . For example, the square next to the top r ight corner shows t he s ta te t hat has h = 22 a t

depth= 12. The gray t one in each area r epresents the number of nodes a t t hat s ta t e

(black=none, white= man y).

0 5 10 15 20

0

2

4

6

8

10

12

H-hat-value

Depth

Figure 4.1.2 Search order of Figure 4.1.1White ar rows in this figure show sta tes which

have same (h + g )-value. In th e IDA* search algorith m, thresholds are set in each iterat ion,

then sea rch is done up to the threshold value. The search progresses from the leftmost arr ow

to right a rrows in successive itera tions.


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Figur e 4.1.3 is th e sam e search a s in figures 4.1.1 a nd 4.1.2 but t he plot sh ows

only a ctua lly sea rched nodes by using t he h function for prun ing 18. The pat h tha t

reached the f irst solution is shown by a ja gged white a rrow run ning from bott om right

to top left . In th is example, tw o a ctions tha t increa sed the h -va lue were needed to

reach the solution. These two actions are represented by the ja gged pa rts of the a rrow,

a nd correspond to thr ee itera tions in ID A* sea rch wit h t hreshold va lues 16, 18, an d 20.In I DA* search, the t hreshold value w a s set t o 16 first beca use the h-va lue of th e initia l

condit ion is 16. The successors th a t ha d bigger ( h+ g)-va lues were not expan ded in th e

first search iterat ion. Only nodes on t he diagonal l ine from h = 16 (nodes h = 15/d= 1 a nd

h= 14/d= 2) w ere sear ched in the first itera tion. In t he second itera tion, the thr eshold

wa s set to 18 a nd ID A* sea rched to the nodes on th e next dia gona l l ine. In t he third

iterat ion, the goa l node wa s found a t d= 20 a nd h = 0.

H-hat-value

Dep

th

0 5 10 15 20

0

5

10

15

20

Figure 4.1.3 Example of Depth-Heuristic Function Visualization (from the Eight P uzzle,Man ha tt an dist an ce wit h blank space) The same insta nce of depth-heuristic function shown in

figures 4.1.1 and 4.1.2, but nodes actua lly searched are shown in th is figure. Note tha t th e

nodes in the upper right of figure 4.1.1 ar e pruned an d not sear ched. The wh ite ar row shows a

pat h to one solution found in this sear ch.

Note tha t t he nodes a bove the diagonal l ine have higher ( h+ g)-values th a n t heth reshold va lue of th e la st itera tion. The nodes shown a bove th e line w ere expand ed in

th e la st it era tion of ID A* sea rch, wh ich recognized th a t t hese successors need not to be

searched. In th is representa tion, we regard th ese nodes as searched, a nd they a re

plott ed in the sa me figure beca use they w ere actua lly expanded.

It is intuit ive to observe the progress of th e sea rch a s a seq uence of depth -

heurist ic function representa tions. Figure 4.1.4 is genera ted from the sam e inst a nce of

18 Note tha t t he scale of the Y-axis is different fr om figure 4.1.1 and 4.1.2, which ma kes the slope of the

search horizon line different. The slopes would be the sa me if we plotted them on t he sam e scale.


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th e IDA* sear ch show n in th e previous figures. Ea ch picture corresponds to one

iterat ion of IDA* sea rch. In t he f irst i tera tion, in th e leftmost picture, only a sma ll

port ion of th e spa ce is sea rched. In t he second itera tion, the algorith m is allow ed to

sear ch the next dia gonal line shown in figure 4.1.2 but does not rea ch the goa l. The goa l

is reached in th e third i tera tion, shown in the rightm ost picture.

Figure 4.1.4 Time Trace of Depth-Heuristic Function Visualization Ea ch figure is asna pshot of an it erat ion of IDA* search in th e depth-heuristic function representa tion. The

leftmost figure is from the first itera tion, the center from the second, the right from the third.

We ca n rega rd figure 4.1.1 a s a s ear ch space representa tion from the view of th e

heurist ic function used for th is problem solving, a nd figur e 4.1.3 as t he result of

pruning . The checkerboard pa tt ern shown in these pictu res results from the heuristic

function used t o map th e sta te in t he problem space into heuristic values. Other

heuristic functions ma y ma p sta tes in the search space into different va lues, an d th us

into different pat terns in depth-heuristic function representa tion. In other w ords, this

representa tion reflects both the st ructure of the sear ch space an d t he heuristic function.Figure 4.1.5 is another example from E ight P uzzle search, using th e sam e initial

condit ion but a slightly different h euristic function19 to estima te the dista nce to the goa l

sta te. The sea rch spa ce ha s basica lly the sam e shape as wit h the previous h function,

but because t he values of successor sta tes can be minus tw o, zero or plus t w o from the

values of the current sta te, h can h a ve only even va lues a nd t he sta te space becomes

evenly spa ced vert ica l lines.

19 In t his example, the h-hat funct ion used ma nhat ta n dis ta nce but didn't include the blank space in the

count.


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H-hat-value

Depth

0 5 10 15 20 25

0

2

4

6

8

10

12

Figure 4.1.5 Example of Depth-Heuristic Function Visualization (from the Eight P uzzle,Man ha tt an d ista nce without the blan k space) All of the nodes up to depth= 12 in the search

tree are shown. The shape of the distribut ion of nodes is an inverted tria ngle similar theprevious search. The striping is a product of the h function used.

Figure 4.1.6 shows nodes a ctually sea rched w ith the h euristic function sh own in

figur e 4.1.5. This figur e corr esponds to figur e 4.1.3. The pat h to rea ch the first solut ion

is shown by the white diagonal arrow.

H-hat-value

Depth

0 5 10 15 20

0

5

10

15

20

Figure 4.1.6 Example of Depth-Heuristic Function Visualization (from the Eight P uzzle,Man ha tt an dista nce wit hout the blank space) This figure shows the nodes actually expanded

in the search, which uses the h function shown in figure 4.1.5 for pruning.

B y compa ring figure 4.1.3 a nd 4.1.6, we can observe th a t t he searched pa th in

figure 4.1.6 is much wider th a n in figure 4.1.3. In t he ideal ca se, a n h function should

return t he exact estima tion of the dista nce to the goa l. Thus th e pa th t o the goal should

be a st ra ight l ine from t he bott om right to the t op left corner, meaning t he a rea


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searched is minimal. The bett er th eh fun cti on, th e nar r ower th e ar ea shown i n t hese

fi gur es, in gener al . In f igure 4.1.6, ten a ctions t ha t increase h values ar e needed to

reach th e goa l sta te. These actions a re represented byvertical a nd right -lea ning line

segments in t he whit e arr ow. The goa l wa s found in t he eighth iterat ion of IDA* sea rch

with this h function. Also th e number of nodes searched w ith t his function is a bout

tw ice as ma ny a s with t he previous h function (figure 4.1.3).

An a nima ted represent a tion of this sear ch, corresponding t o figure 4.1.4, is

shown in figure 4.1.7. As it took eight itera tions to rea ch the goa l with t his h function,

th ere a re eight pictur es in th is figure. The first itera tion is show n in the top left corner,

th e second in t he top second left, a nd so on.

Figure 4.1.7 Time Trace of Depth-Heuristic Function Visualization (from the Eight P uz-zle, Man ha tt an dist an ce wit hout the blank space) The same representat ion as figure 4.1.4 but

wit h a different sear ch algorithm. These figures are ordered from left to right an d top to bottom.

As th e sea rch progresses, the frontier of th e sea rch moves towa rd t he top left

corner, while at the sa me time, the tra ce becomes wider. P rogress is significa nt in t he

last t wo i terat ions , suggest ing tha t t he h function w orks better a t deeper levels tha n in

sha llow levels.


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5. Time Series Representations

5.1 Time Domain Representations

When w e record search depth a s th e history of sear ch, we ca n observe the beha v-

ior of search in the t ime doma in, and w e ca n a pply time domain a na lysis techniques.

Figur e 5.1.1 is an exa mple from a time t ra ce of genera l depth-first sea rch. The

X-a xis is th e order of sea rch, wh ich corresponds t o time if we a ssume const a nt t ime to

expa nd ea ch node. The Y-a xis is the depth of the sea rched node. In t his exa mple, th e

a lgorithm searched down to depth = 7 and continued visiting nodes at depth 6 and 7

several hundred times. B ig spikes in this representa tion mea n big backtra cks. For

example, a ba cktra ck from depth 7 to depth 3 ha ppened at t ime 130. Time 0 th rough

time 130 corresponds t o a sear ch of one bra nch from level 3 if th e sear ch is done in

depth-first order.

In the t ime tra ce of pure depth-first search, th e dista nce between points in the

series at one level represent s time t o determ ine one subtr ee below th a t level.

Depth

Time

Figure 5.1.1 Time Trace of Depth-first Search The X- an d Y- axes represent t ime a nd t he

depth visited at tha t t ime. In depth-first search, algorithm picks up one branch an d tries to

visit the deepest level of the bra nch first , then goes back and forth a t th e deeper levels. The

big spikes in this representa tion signifies the termina tion of searching a subtr ee, wit h a la rge

backtra ck to the next subtree.

I t is ea sy to see a t w hich levels the a lgorithms spent t ime. For example, figure

5.1.2 represents t he time tr a ce of brea dt h-first sea rch on t he sa me search spa ce a s in

figure 5.1.1. B rea dt h-first sear ch visits all nodes a t sha llow er levels first . The time

spent a t a level is proportiona l to th e number of nodes a t tha t level, so the sha pe of th e

time trace is completely different between depth-first search and breadth-first search.

Depth-first a lgorithms h a ve spiky time tra ces a nd breadt h-first a lgorithms m a ke stairs.


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Time

D

epth

Figure 5.1.2 Time Trace of Breadth-first Search

However, it is n ot so easy t o f ind pa tt erns in time tr a ces for a lgorithms other

th a n pure brea dt h-first sea rch and depth -first sear ch. For exa mple, figure 5.1.3 is th e

time tr a ce of ID A* sea rch for th e problem discussed in the previous sections. It is very

ha rd t o tell much from th is f igure.

0 50 100 150 200 250 300

5

10

15

20

5

Figure 5.1.3 Time Trace of IDA* Search on Eight Puzzle

From the time tra ce of search a lgorithms w e ca n get: 1) a rough idea of the

beha vior of the sear ch a lgorith ms; it is especially useful for pure depth -first sea rch or

bread th -first sea rch, a nd 2) some idea of th e frequency component s of time t ra ce; this is

needed for th e design of a smooth ing filter (described t he following section).


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5.2 Frequency Domain

Time domain a na lysis is stra ightforwa rd a nd very intu itive. Although it cann ot

distinguish tw o similar a lgorithms in deta il , i t is useful to see the search depth h istory.

As t he next step, we can a pply fourier t ra nsforms t o view t he time tr a ce of sea rch

a lgorithms in t he frequency doma in. B y converting a sequence from th e time doma in

into the frequency doma in, we can observe itera tive actions in th e search a lgorithms

more clea rly. For exa mple, visiting a n ode a t depth 5, expa nding t hree child nodes a t

depth 6 and then r eturning t o a node at depth 5 corresponds a n itera tion of interval 4.

If t his ha ppens often, it ca n produce a st rong frequency component a t fr equency 1/4 Hz.

Frequ ency components represent ba cktra cking in sear ch. In genera l, low frequency

components represent large backtracks, which are likely to occur when returning to a

sha llow level of th e sear ch tree, a nd h igh frequ ency component s come from minor

backtracking at deeper levels.

It era tive components observed in th e frequency doma in can be ca used by

intera ctions between the structure of the search space a nd search a lgorithms. For

exa mple, th e time tr a ce of depth-first sea rch a nd brea dth -first sea rch can include

completely different frequency components even w hen a pplied to exactly t he sa me

search space. Thus, frequency a na lysis ca nnot separa te th e structure of the search

space from chara cteristics of sear ch a lgorithms.

In t his subsection, the bas is of the fourier tr a nsform is explained first. Then a

smooth ing technique for pra ctical ana lysis is reviewed. To illustra te frequency a na lysis

of search algorithms, we show frequency gra phs for th e Eight P uzzle a nd t he tra veling

salesperson problem, w ith their int erpreta tions.

5.2.1 Fourier TransformTh e four ier t r ansformis a common technique to tra nsform a function in t he time

domain into the frequency doma in. The basic idea is to regard a n a rbitra ry w a ve form

a s the ad dition of simple sine wa ves. The tra nsform decomposes the wa ve into sine

wa ves, yielding the strength a nd pha se of ea ch decomposed sine wa ve. B y a dding all of

th e decomposed sine wa ves, w e ca n reconst ruct t he origina l wa ve form. This is ca lled

t he in ver se fouri er tr ansform.

In the ca se wh ere the origina l function is continuous, th e following formula s a re

a pplied t o ca lculat e the fourier t ra nsform a nd inverse fourier tra nsform.

F ( ) = f t( ) e2it

dt FourierTransfor

f t( ) = F ( ) e2it

d InverseFourierTransfor

(5.2.1)


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If t he original fun ction is in d iscrete form, equa tions (5.2.2) a re used t o ca lculat e

the fourier tra nsform a nd inverse fourier tra nsform. They ar e basica lly equivalent t o

th e equa tions in (5.2.1) but a ssume a discrete number of wa ve forms.

F j( ) =1

N

f t( ) e2ikj/T

k=0

N1

FourierTransfor

f k( ) = F j( ) e2ikj/Tj=0

N1

InverseFourierTransfor

(5.2.2)

Note tha t in t he case of th e discrete fourier tra nsform, th e highest frequency

depends upon t he sam ple ra te of the original function. I f th e sampling ra te is 1 point

per second, th e highest fr equency is 1/2Hz. For a sa mpling ra te of 10 point s per second,

we can get a highest frequency of 5Hz 20.

As we a re going to a pply the fourier tr a nsform to time tr a ces of search

a lgorith ms, we need to use th e discrete fourier tr a nsform. Figur e 5.2.1 shows a

frequency domain spectrum from the node-depth h istory of a depth-first sea rch.

Index

Strength

Figure 5.2.1 Frequency Distribution of depth-first search in figure 5.1.1

In t his frequency distr ibution gra ph, th e X-a xis represents frequ ency (or

interval) a nd th e Y-a xis represents relat ive strength . B y simply a pplying equat ion

(5.2.2) to t he discrete t ime series of sea rch, th e frequency components a re calcula ted a s

complex num bers. In t his gra ph, the absolute values of each frequency component a recalculat ed f irst , t hen they a re sta nda rdized by dividing by th e maximum component

va lue. Note also tha t t he zero frequency, which is referred t o a s D.C. level in electr ica l

engineering, is omitted because it represents simply the average node depth.

20 Sa mpling theory s ta tes tha t t he theoret ica l highest frequency is ha lf of the sam pling ra t e.


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In t his example, 320 da ta point s ar e used for th e ca lcula tion. Thus 160 in th e X-

a xis corr esponds t o 160/320 = 0.5 Hz or int erva l 2. Sim ila rly 107 in t he midd le of th e X-

a xis corr esponds t o itera tions a bout in t erva l 320/107 = 3, 80 is from int erva l 320/80 = 4,

a nd so on. Int erva l 4 is ca used by the itera tive sea rch of 3 child nodes. For exa mple,

depth-first sear ch from a level 4 node follow ed by th ree lea f nodes, which is t he

sequence of 4,5,5,5,4,5,5,5,. . . in th e time t ra ce, can cause fr equency components a t th einterval 4. In general in depth-first search, frequency components a t int erval i

correspond to search on the node wh ich ha s i -1child nodes.

As th is exa mple ca me from discrete numbers, th e oth er interva l component s

betw een th ese va lues cann ot technica lly exist, th ough they ma y exist in practice. They

a re called harmonicsa nd a re ca used by decomposing non-sine wa ves int o sine wa ves in

the fourier tra nsform.

The frequency distribut ion show n in figure 5.2.3 w a s genera ted from th e time

tra ce of depth-first search on a fully ba lan ced t ree with b= 3. There is a big clear peak a t

a bout 75, wh ich corr esponds t o inter va l 310/75 4 in this distribution. I f the searchtrees are ba lanced an d ha ve exactly th e same bra nching factors in each level, frequencydistributions t end to ha ve clea r peaks, though th ey sti l l have ha rmonics.

Index

Strength

Figure 5.2.3 Frequency Distribution of Depth-first Search on a Fully Balanced Tree

5.2.2 Smoothing Filter

In t he fourier a na lysis of real sea rch histories, noise in th e origina l functionma kes a na lysis difficult. We can a pply a smoothing filter to th e frequency distribut ion

to reduce noise and to make the distribution clear.

Although w e ca n design a ny f i l ter in pra ctice, the most common is the mea n

filter wit h a fixed wind ow size. It w orks as a low-pa ss (high-cut) filter beca use the noise

is often in th e high ran ge of frequency distribut ions. E qua tion (5.2.3) is used as th e

simple mean filter of window size 2.


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MeanSmoothed a i( )[ ] =a i( ) + a i+1( )

2(5.2.3)

In general , th e mean smoothing f i l ter with window size wis described by t he

follow ing formula :

MeanSmoothed a i( ),w[ ] =1

wa i( )

k=iw/2

i+w/2

(5.2.4)

Note here tha t a pplying a mean fi lter w ith w indow size 2 twice to a function f is

equivalent t o applying a qua rter-ha lf-qua rter f i l ter to th e function fonce.

We ca n a lso design a high-pa ss filter or other filters, but for smooth ing purposes,

only low -pas s filters ma ke sense.21 We should look for the filter t ha t m ost clea rly

reveals t he frequency components.

I t is importa nt tha t smoothing f i lters not be applied to time domain da ta (beforefourier tra nsform), but ra ther t o the frequency doma in (a fter fourier tra nsform). I f

a pplied to the time doma in, a filter reduces high frequency components of the origina l

w a ve form. As a result , the frequency distr ibution would be different from th e original

form. Applying a low pa ss filter to the time domain is equivalent t o ignoring the high

frequencies in th e frequency domain. B eca use we are interested in tr end in the

frequency domain, we should apply the smoothing filter to the frequency distribution.

The following gra ph w a s generat ed by a pplying a mean smoothing f i lter four

times to the frequency dist ribution shown in figure 5.2.1. Note tha t spikes in the

unsmooth ed distribution ha ve been reduced by smoothing so tha t pa tt erns in th e

frequency spectr um become clear er. To get a n idea w hich frequency component s rea llydo exist a nd w hich do not, w e need to check the figures from the time doma in.

B y smoothing, frequency components a t intervals 3 a nd 4, located a t index 107

a nd 80 respectively, ha ve been enha nced. How ever, it is still difficult t o distin guish

other components betw een these levels th a t a re ca used by ha rmonics of low fr equency

components.

Note here tha t the smoothing f i lter must be designed so tha t i t doesn't f i lter out

components w hich interest us. Thus designing t hese filters requires knowledge of the

frequency components t ha t sh ould exist in t he observa tions. We need to check the time

tra ce of the search f irst in order to get some idea wh ich pa rt of th e frequency

components a re importa nt.

21 We can use t he sam e fourier t echniqu e to check the freq uency response of any filter. This is th e most

common usa ge of the fourier tra nsform in electrical eng ineering.


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Index

Strength

Figure 5.2.3 Smoothed frequency distribution of depth-first search with mean filterA smoothing filter is a pplied to t he frequency dist ribution shown in figure 5.2.1. The smoothing

filter can r educe spikes, revealing the m ain components of the distr ibution.

5.2.3 Examples from Fourier Analysis of Search Algorithms

A Simple Example of Fourier Transform on Search

The following figure was generated from the history of a breadth-first search.

We should see no ba cktra cking in t his tr a ce beca use th e algorithm s can s a ll nodes from

sha llow er to deeper levels once a nd never visits t he sa me level a ga in. Therefore, there

should be no frequency component s in the frequency doma in. In pra ctical applica tions

of the fourier tra nsform, i t is assum ed tha t t he wa ve form is sampled from an infinite

sequence, and furt hermore, tha t t he rest of the w a ve is exa ctly the sam e as t he observedpar t. This ass umpt ion causes us to find low frequency components even th ough the

original function ha s no itera tive component s. As a result, the frequency distr ibution

ha s only low fr equency component s, wh ich a re basica lly noise.

This result should be contra sted t o the depth -first s ear ch shown in figure 5.2.5.

Here, lots of backtr a cking cau ses high-frequency components in t he frequency doma in.

Therefore, th e frequency dist ribut ion of depth-first sear ch includes more high frequen cy

components than breadth-first search. In general, high frequency components

correspond to ba cktra cking w ith a short period, which can only occur in t he low est levels

of a s ear ch tree. Low frequency components correspond to long period ba cktra cking,

w hich can occur betw een high levels of sear ch trees.


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Index

Strength

Figure 5.2.4 Frequency Distribution of Breadth-first Search22 Only low frequencycomponents exit in this d istribution, w hich is called itera tive noise. (This phenomenon results

from using a finite number of sam ple points.)

Frequency Distribution of Combined SearchTo demonstr a te t he utility of the fourier tra nsform, th e figures in 5.2.5 a re

creat ed from combined-sear ch algorithm s. The Br eat h-Depth a lgorith m, shown a s (A),

uses brea dth -first search unt il it r eaches depth 4, then uses depth-first search. On th e

contr a ry, t he Depth-B readt h a lgorithm in (B ) uses depth-first sea rch unt il level 4, a nd

then u ses brea dth-first sear ch.

50 100 150 200 250 300

2

3

4

5

6

7

50 100 150 200 250 300

2

3

4

5

6

7

Dept h Dept h

Time Time

(A) B rea dt h-Dept h S ea rch (B ) D ept h-B rea dt h S ea rch

Figure 5.2.5 Time Trace of Breadth-Depth Search and Depth-Breadth Search Timetra ce of two combined search algorithm s. The Br eat h-Depth a lgorithm, shown a s (A), uses

breadt h-first search until it reaches depth 4, then uses depth-first search. In contr ast , the

Depth-Brea dth a lgorithm in (B) uses depth-first search unt il level 4, then bread th-first sear ch.

22 A mean sm ooth filter wa s applied four times to generat e this graph.


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In the t ime doma in, i t is dif f icult to compare t wo t ime tra ces of sea rch depth

directly . Even th ough we ca n calcula te th e correlat ion of two time tra ces by cha nging

th e offset of one curve, we n eed to compa re t his correla tion for each possibl e offsetin

order to find th e best va lue. For exa mple, ima gine the case shown in figure 5.2.6. The

sam e algorithm is used in tw o almost identical sear ch trees; one search tr ee has depth =

4 (shown in A) a nd t he other a lso ha s depth = 4 but one extra bra nch at depth 2 (shownin B). B eca use of the extra bran ch, the time tra ce of the second search tree ha s a delay,

thus w e will f ind the best correlation with a n unknown offset. As we don't ha ve any

idea a bout t his offset va lue in general , we need to itera te this calculation man y times.

This sit ua tion w ill become even more complicat ed w hen w e compare t ime tra ces from

dissimilar trees.

(A) (B)

?

1,2,3,4,5,5,4,5,5,... 1,?,?,?,?,...,2,3,4,5,5,4,5,5,...

offset identical with (A)

SEARCHT

REE

TIME

TRACE

1

3

1

2

3

4 4

5 5

Figure 5.2.6 Problem of the correlation of time traces (B ) ha s almost identical search

tree with (A) but ha s an extra bra nch at level 2. As a result of this extra bra nch, the timetra ces must be offset to find the best correlation.

Once time traces are converted into frequency distributions they are directly

compa ra ble a ma jor benefit for t he use of frequency a na lysis.

B y using th e frequency distribution representa tion we can easily discuss

bread th -first a nd depth-first components of th ese tw o sear ch a lgorith ms. The

distr ibutions a re shown in figure 5.2.7.

The frequency dist ribut ion of Depth-B rea dt h sea rch show n in figur e 5.2.7(A) ha s

strong spikes at the sa me pla ces a s th e pure depth-first search a lgorithms shown in

figure 5.2.2. Thus w e ca n predict a simila rity betw een D epth-B readt h sea rch a nd puredepth-first sear ch. The frequency distribut ion of B rea dt h-Depth sea rch, shown a s figure

5.2.7(B ), ha s only low frequency components a nd is simila r t o pure brea dt h-first sea rch

shown in figur e 5.2.3.


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0 25 50 75 100 125 150

0.2

0.4

0.6

0.8

0 25 50 75 100 125 150

0.2

0.4

0.6

0.8

1St rength St rength

Index Index

(A) B rea dt h-D ept h Sea rch (B ) D ept h-B rea dt h Sea rch

Figure 5.2.7 Frequency Distributions of Breadth-Depth Search and Depth-BreadthSearch Frequencies included in th e Br eadt h-Depth sea rch in (A) include many more highfrequency components, which w ere generat ed by depth-first components. The distribut ion

from the Depth-Brea dth sea rch in (B ) ha s only low components. These are basically n oisecaused by errors of tra nsformat ion and ha rmonics. This distribution is very similar to those

produced by pure breadth -first search a lgorithms.

Frequency Distribution of IDA* Search on Eight PuzzleAs a nother exa mple, the follow ing figures represent fourier tra nsforms of IDA*

search on th e Eight P uzzle. A mean smoothing f il ter wa s applied four times in the

frequency domain.

0.00

0.10

0.20

0.30

0.40

0.00 0.10 0.20 0.30 0.40 0.50

Frequency

Strength

0.00

0.10

0.20

0.30

0.40

0.00 0.10 0.20 0.30 0.40 0.50

Frequency

Strength

Figure 5.2.8 Frequency Distribution of IDA* Search on the Eight Puzzle (A) wit hManha t ta n dis ta nce (B) with count f ina l posit ion

Looking at th ese two figures w e ca n compar e the behavior of th e tw o heuristic

functions. The frequency distribution from IDA* search w ith Ma nha tt a n dista nce (A)

only includes lower frequencies, especially w hen compa red w ith fina l position count (B ).


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This indicat es less ba cktra cking in lower levels of th e tree. The sear ch works by going

down t o the deeper levels, th en does a lit tle backtr a cking followed by a r etur n to higher

levels. The heurist ic function helps by prunin g bad br a nches in early st a ges of the

search, and a s a r esult , ma ny big ba cktra cking jumps are observed. On the contr a ry, in

th e ca se of th e count fina l position function, shown in (B), th e frequency distribut ion

includes relat ively ma ny high frequency component s. This mean s more backtra ckingoccurs in both high levels a nd low levels, from w hich we can conclude tha t t he heuristic

function didn't help much to prune bra nches.

Figur e 5.2.9 is th e frequency distribut ion of pure depth -first sea rch on the E ight

P uzzle. The num ber of successors is either 1, 2, or 3 an d depends on the current sta te,

th us the bra nching factor of th e puzzle is not const a nt . Therefore, the distribut ion is

more sca tt ered t ha n t he distributions in f igure 5.2.8.

0.00

0.10

0.20

0.30

0.40

0.00 0.10 0.20 0.30 0.40 0.50

Strength

Frequency

Figure 5.2.9 Frequency Distribution of Pure Depth-first Search on the Eight PuzzleThere ar e strong peaks a t bra nches of 3, 4 an d 5.

5.3 Interval Counting Methods

The interval count ing representa tion is an a lterna tive wa y to view the interva ls

of wa ve forms. B eca use the fourier ana lysis of t ime tra ces ha s the problem of

ha rmonics, we can not get a clear picture from fourier an a lysis without some knowledge

a bout w hich frequencies to use in a smoothing f i l ter . In contr a st, th e int erval count ing

method ca n be used with out a ny knowledge of frequencies. U sing numbers, this methodcombines the representation of sub-tree and branching factors from frequency analysis

(th ough it only w orks if the sea rch is done in depth-first order).

Suppose we ha ve a t ime tra ce of search depth in some algorithm shown a s f igure

5.3.1. We can mea sure the interva ls betw een occurrences of th e sa me search depth. For

exa mple, depth 4 appea rs four times in this search a nd t hese interva ls are 2, 1 a nd 1.


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We ca n count the int ervals between different search depths a nd f i l l out a ta ble, as

shown in t a ble 5.3.1.

1 2 3 4 5 4 4 4 3 2 1tttt ++++

Figure 5.3.1 Interval Counting Method The numbers on th e arr ow a re depths of nodesvisited. To calculat e the int erval for each level, it is necessary t o find the n ext occurrence of a

node at t ha t level in the sequence, and count the n umber of nodes from the initia l occurrence to

the next. For example, 2 is found a t t= 2 and t= 10, thus the interva l of 2 is 8.

Table 5.3.1 Example of Interval Counting Table

In t erva l D ept h = 1 D ept h = 2 D ept h = 3 D ept h = 4 D ept h = 5 D ept h = 6

2 2

3 1

4

5

6

7 1

8

9 1

10

In t he fourier tra nsform, the interva ls are count ed betw een a ll pa irs of the sa me

depth, w hich makes it difficult to interpret periodicity in th eir beha viors. For exa mple,

for d epth 4 in figure 5.3.1, we w ill find frequ ency components of interva l 2 (tw ice), 3

(tw ice), 4 (once), an d 5 (once) from t he fourier tr a nsform . The only int erest ing freq uency

components in t his list a re neighboring levels beca use we a re interest ed in the sma llest

intervals a nd interva l 5 isn't one of the sma llest int ervals of depth 4 here. Only interval2 (tw ice) a nd 3 (once) should be count ed, a nd th e other component s sh ould be ignored,

a s show n in figur e 5.3.2.


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1 2 3 4 5 4 4 4 3 2 1tttt ++++

2

3

45

frequencycomponents

for depth=4

Figure 5.3.2 Frequency Components for depth=4 An an alogy to the fourier tra nsform isfound in interva l counting. In a fourier tra nsform, all combinat ions of the interva l shown in this

figure would be counted, which would obscure ra ther t ha n reveal th e frequency components.

The follow ing t w o figures a re examples of int erva l count ing from depth -first

search a nd breadt h-first search, graphed a s thr ee-dimensional bar -plots. In t hese

figures, the X-a xis represent s th e interva l, the Y-a xis represent s th e level and th e Z-a xis

represents frequency of a ppea ra nces in the histories of the sear ch. Note th a t aloga rith mic function wa s applied to frequencies to ma ke sma ll components visible on

the sa me sca le with t he bigger values, wh ich a re of dif ferent ma gnitude.

5

10

15

20

2

4

6

0

2

4

5

10

15

2

4

6

0

2

4

5

10

15

20

2

4

6

0

2

4

5

10

15

2

4

6

0

2

4

Level

Interval

Frequency

Level

Interval

Frequency

(a) Depth-first Search (b) Breadth-first Search

Figure 5.3.3 Three Dimensional Bar Plot of (a) Depth-first Search, (b) Breadth-firstSearch in a randomly pruned tree In th ese gra phs, only the counts w hich are less than 20interva ls are shown . There are lots of long interval components in t he depth-first sear ch

shown a s (a).23 On the contra ry, in the brea dth-first sea rch gra ph, only components which

ha ve an int erval of 2 exist .

23 Note tha t counts sma ller than d epth 3 in depth-first search ha ve very big interva l s and a re located in

the fa r right positions of this plot.


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In d epth-first search, the interva l a t a certa in level represents t he number of

nodes below t ha t d epth. For exam ple, if we observe level 2 follow ed by 30 lower levels,

a nd then level 2 a ga in, tha t mean s the bra nch has 30 nodes total . In th e interval

count ing method, this is counted at level = 2 a nd interva l = 31. Similarly , the

observation of level = dand interval = isuggest s th e exista nce of one subtree at level d

which ha s i -1nodes in the subtree. I f the tree is bala nced, we ca n expect t ha t t he totalnu mber of nodes below level i is b

hi , where h is the height of the tree and b is the

bra nching fa ctor of the subtr ee. It is a lso expected tha t w e will observe level = da nd

interval = i, bi

t imes in a fully balan ced tree with th e same pa ra meters.

In brea dth-first search, we see no depth other th a n interva l 2 using interval

count ing because brea dt h-first s ear ch progresses level by level, yielding only int erva ls of

2. In t his gra ph, the heights of bars of interva l 2 represent exactly the sa me number of

nodes at each level.

The follow ing figures show int erva l counting of th e time tra ces for the sear ch

a lgorith ms d escribed in section 5.1.

B rea dt h-Depth -first sea rch in figure 5.3.4(a ) uses bread th -first sea rch down t o

level 3, and t hen sw itches to depth-first sea rch. Thus t he interva l counting gra ph

becomes exa ctly th e sa me gra ph a s depth -first sea rch below level 3. Above level 2 it is

the sa me as breadt h-first search an d ha s one bar a t level 2 a nd interva l 2.

5

10

15

20

2

4

6

0

2

4

5

10

15

2

4

6

0

2

4

5

10

15

20

2

4

6

0

2

4

5

10

15

2

4

6

0

2

4

Level

Interval

Frequency

Level

Interval

Frequency

(a) Breadth-Depth-first Search (b) Depth-Breadth-first Search

Figure 5.3.4 Three Dimensional Bar Plot of (a) Breadth-depth-first search, (b) Depth-breadth-first search in a randomly pruned tree


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On t he other ha nd, D epth-B rea dt h-first sea rch in figure 5.3.4(b) la cks the bar of

breadth -first sea rch at int erval 2. I t ha s bars a t interva l 2 a bove level 3 instead , which

represents bread th -first sear ch components. This mean s the search uses brea dt h-first

search methods below level 3. Two sma ll ba rs in a bigger interval a rea w ere generat ed

by counting t he sam e level of a different bra nch. These sma ll components a re regar ded

a s ghost images a nd can be avoided by not count ing an interval in t he same level butdifferent bra nches.

6. Summary of Representations of Search

This section is represented by a summ a ry t a ble w hich is too la rge to include in

th is post script file. It is, however, included in the ha rdcopy version of th is technical

report which can be obtained via e-mail request, [email protected], or

Ba rbara Suther land

Computer Science Dept., Box 34610

U niversity of Massa chusett s

Amherst, MA 01003-4610

(413) 545-3286

7. Conclusions

Theoretica l an a lysis techniques a re bett er tha n experimenta l ana lysis.

Sometimes, i t is very diff icult to build t heoretica l models a nd a na lyze them w hen

problems become complex, or heur istic functions a re used t o solve the problems. In th e

a na lysis of search a lgorithms w ith h euristic functions, i t is very ha rd t o formalize the

problem a nd t o solve it by theoretically even in a sma ll problem like th e Eight P uzzle.

Visualizat ion t ools for t he search a lgorithms a re importa nt in these ca ses. They can

also be applied to debugging search algorithms in practice.

Time doma in a na lysis of search a lgorithms a re shown in thr ee forms:

(a ) time tra ce to see at a glance the behavior of search

(b) frequency domain to a na lyze iterative a ction of search a lgorithms

(c) interva l counting t o decompose sea rch a lgorith ms int o depth a nd brea dt h

components.

Alth ough th ese tools cann ot be directly a pplied to ar bitra ry ca ses, the effort to collect

such tools an d to generalize th em is very importa nt . We need to continue developing

such tools as part of the st udy of algorith ms.


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A. Search Algorithms used in Examples

A.1 Pseudo code of Depth-first, Breadth-first and Best-first Search

/ * Gener al Search Funct i onsear ches a t r ee def i ned by node_l i st and f unct i on successor( ) i n t heor der of st r at egy( ) f uncti on */gener al _sear ch( node_l i st , st r at egy){

/ * Check i f t here i s a node */i f ( empt y( node_l i st ) )

r et urn ( NOT_FOUND) ;

cur r ent _node = f i r st ( node_l i st ) ;al t er nat i ve_nodes = r est ( node_l i st ) ;

/ * Check whether t he cur r ent node i s a goal */i f ( goal p( cur r ent _node) )

r et ur n( cur r ent _node) ;

/ * Expand successor nodes */successor _node = successor s( cur r ent _node) ;

/ * Re- cal cul at e sear ch orderst r ategy( ) f unct i on r eturns t he order of successor nodes */node_or der = st r ategy(al t ernat i ve_nodes, successor_node);

/ * Cal l sel f r ecur si vel y */r et ur n ( gener al _sear ch( node_or der , st r at egy)) ;

}

/ * Depth- f i r st Search

f unct i on dept h_f i r st per f or ms dept h- f i r st sear ch over t he t r ee def i nedby node_l i st and successor ( ) f unct i on*/

ddddeeeepppptttt hhhh____ffffiiii rrrr sssstttt ( node_l i st){

general _sear ch( node_l i st , *dept h_f i r st _or der( ) ) ;}dept h_f i rst _order( node_l i st, successor_l i st){

r et urn(append( successor _l i st , node_l i st ) ) ;}

/ * Br eadth- f i r st Sear chf unct i on br eadt h_f i r st per f or ms br eadt h- f i r st sear ch over t he t r ee

def i ned by node_l i st and successor( ) f unct i on*/bbbbrrrr eeeeaaaaddddtttt hhhh____ffffiiii rrrr sssstttt ( node_l i st){

general _sear ch( node_l i st , *breadt h_f i r st _or der ( ) ) ;}breadt h_f i rst _order( node_l i st, successor_l i st){

r et urn(append( node_l i st , successor_l i st ) ) ;}


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/ * Best- f i rst Searchf unct i on best _f i r st per f or ms best - f i r st sear ch over t he t r ee def i nedby node_l i st and successor ( ) f unct i on

*/bbbbeeeesssstttt____ffffiiii rrrr sssstttt ( node_l i st)

{general _search( node_l i st, *best_f i rst _order( ) ) ;

}best_f i rst _order( node_l i st, successor_l i st){

return( sort _by_pri or i t y( append( node_l i st, successor_l i st) ) ) ;}


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A.2 Pseudo code of A* Search

/ *f unct i on A* per f orms A* search over t he t r ee def i ned by node_l i st andsuccessor ( ) f uncti on*/

aaaa****(((( nnnnooooddddeeee____llll iiii sssstttt )))){

/ * Check i f t here i s a node */i f ( empt y( node_l i st ) )

r et urn ( NOT_FOUND) ;

cur r ent _node = f i r st ( node_l i st ) ;al t er nat i ve_nodes = r est ( node_l i st ) ;

/ * Check whether t he cur r ent node i s a goal */i f ( goal p( cur r ent _node) )

r et ur n( cur r ent _node) ;

/ * Expand successor nodes */successor _node = successor s( cur r ent _node) ;al l _successor s = append( al t ernat i ve_nodes, successor_node);

/ * Re- cal cul at e sear ch order */node_or der = sor t ( al l _successor s, f ( $node) ) ;

/ * Cal l sel f r ecur si vel y */r et ur n (a*( node_order ) ) ;

}

ffff(((( nnnnooooddddeeee)))){

r eturn (g( node) + h(node)) ;}

gggg(((( nnnnooooddddeeee)))){

/ * use dept h f r om r oot as f val ue */r eturn dept h(node) ;

}/ * Heuri st i c Funct i on */hhhh(((( nnnnooooddddeeee)))){

r eturn manhat t an_di st ance(node) ;}


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A.3 Pseudo code of IDA* Search

/ *f unct i on i da* perf orms i da* sear ch unt i l MAX_TH over t he t r ee def i nedby r oot _node and successor ( ) f unct i on*/iiii ddddaaaa****(((( rrrr ooooooootttt____nnnnooooddddeeee))))

{/ * i ni t i al t hreshol d */t h = f ( r oot _node);

/ * i d sear ch */l oop {

smal l est _f = +I NFI NI TY;t h = df i da*( r oot _node, t h) ;

} unt i l ( goal p( t h) | | t h > MAX_TH) ;

i f (goal p( t h) )return (th);

el ser et urn ( NOT_FOUND) ;

}

/ * Thi s gl obal v

mm-visualization tools for real-time search algorithms

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