shortest paths - university of texas at san antoniobylander/cs5633/notes/shortest-paths.pdfall-pairs...

CS 5633 Analysis of Algorithms Chapter 24 and 25: Slide – 1

Shortest Paths

Single-Source Shortest PathsDijkstra’s AlgorithmBellman-Ford AlgorithmDifference ConstraintsAll-Pairs Shortest PathsFloyd-Warshall AlgorithmJohnson’s Algorithm

Shortest-Paths

⊲ shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Shortest path problems on weighted graphs(directed or undirected) have two main types:

Single-Source Shortest-Path: find the shortestpaths from source vertex s to all other vertices.

All-Pairs Shortest-Path: find the shortest pathsbetween all pairs of vertices.

Some notation:w(u, v) = weight of edge (u, v)w(p) = sum of weights on path p

Let δ(u, v) = shortest distance from u to v.

Let sopt❀ v stand for shortest path from s to v.

Example Weighted Graphs

shortest paths

⊲ example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


1

23

7

85

54 7

9

6

4

8

a

b c d

ei

h g f

Example Bad Graph

shortest paths

example graphs

⊲ bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Graphs with negative-weight cycles can’t besolved.

Key Properties

shortest paths

example graphs

bad graph

⊲ properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Triangle Inequality: If (u, v) is an edge, thenδ(s, v) ≤ δ(s, u) + w(u, v). If p is a path from u

to v, then δ(s, v) ≤ δ(s, u) + w(p).

Proof: δ(s, v) > δ(s, u) + w(u, v) contradicts δ’sdefinition. So does δ(s, v) > δ(s, u) + w(p).

Optimal Subpath Property: If u is on sopt❀ v, then

δ(s, v) = δ(s, u) + δ(u, v).

Proof: Let p be sopt❀ v, and let p1 and p2 be the

subpaths s ❀ u and u ❀ v. By definition,w(p) = w(p1)+w(p2) ≥ δ(s, u)+ δ(u, v). A lowerδ(s, u) or δ(s, v) would contradict p’s optimality.

Key Properties Continued

shortest paths

example graphs

bad graph

properties 1

⊲ properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Convergence Property: If s ❀ u→ v is a shortestpath, then δ(s, v) = δ(s, u) + w(u, v).

Proof: Follows from the optimal subpath property.

Let v.d ≥ δ(s, v) for all v with s.d = 0 = δ(s, s).

Path-Relaxation Property: If v.d > δ(s, v), thensome edge (x, y) satisfies v.d = δ(s, x) andy.d > x.d+ w(x, y).

Proof: Some edge (x, y) on sopt❀ v must be the

first edge with x.d = δ(s, x) andy.d > x.d+ w(x, y) = δ(s, y).

Basic Procedures

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properties 1

properties 2

⊲basic procedures

1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


These properties justify the following proceduresfor single-source shortest-paths.v.d = distance from s to v.v.π = previous vertex on s ❀ v path.

Initialize-Single-Source(G, s)for each vertex v in G

v.d ← ∞v.π ← nil

s.d ← 0

Initial ∞ ensures that d.v ≥ δ(s, v) for all v.s.d = 0 ensures that s.d = δ(s, s).

Basic Procedures Continued

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

⊲basic procedures

2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Relax(u, v, w)if v.d > u.d+ w(u, v)v.d ← u.d+ w(u, v)v.π ← u

If u.d ≥ δ(s, u) and v.d ≥ δ(s, v) beforeRelax(u, v, w) then they remain true afterwards.

Also, if v.d > δ(s, v), then the Path RelaxationProperty implies that some call to Relax willimprove d somewhere.

Dijkstra’s Algorithm

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

⊲ dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Dijkstra’s Algorithm assumes all weights arenonnegative.

Dijkstra(G,w, s)Initialize-Single-Source(G, s)S ← ∅Q ← G.V using d field for priority queuewhile Q 6= ∅

u ← Extract-Min(Q)S ← S ∪ {u}for each v ∈ G.Adj[u]

Relax(u, v, w)

Dijkstra Analysis

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

⊲ dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Proof of Correctness:

Basis: s is assigned to u, s.d = 0 = δ(s, s)Assume: For all x ∈ S, x.d = δ(s, x)Show: u.d = δ(s, u) when u is extracted.Induction: Proof by contradiction.

Suppose δ(s, u) < u.d.

Some edge (x, y) in sopt❀ u goes from S to Q.

y.d = δ(s, y) because (x, y) has been relaxed,but u.d ≤ y.d = δ(s, y) ≤ δ(s, u) < u.d

is a contradiction.

Dijkstra is O(E lg V ) with binary heaps.

Bellman-Ford Algorithm

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

⊲ bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


This works with negative weights. false isreturned if a negative cycle is detected.

Bellman-Ford(G,w, s)Initialize-Single-Source(G, s)for i ← 1 to |V | − 1

for each edge (u, v) in G

Relax(u, v, w)for each edge (u, v) in G

if v.d > u.d+ w(u, v)return false

return true

Bellman-Ford Algorithm

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

⊲ bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Proof of Correctness:

All finite shortest paths have ≤ V − 1 edges.

The ith iteration relaxes ith edge in the optimalpath.

If shortest path is finite, then after V − 1iterations, v.d = δ(s, v).

If there is an infinite shortest path,then v.d > u.d+ w(u, v) for some edge (u, v).

Bellman-Ford is O(V E).

Difference Constraints

shortest paths

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properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

⊲ diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


A difference constraint is xj − xi ≤ bk, where xiand xj are variables and bk is a constant.

Difference constraints can represent timerelationships between events.

If x2 must occur within 2 hours after x1, thenx2 − x1 ≤ 2.

If x2 must occur at least 2 hours after x1, thenx2 − x1 ≥ 2, which is equivalent to x1 − x2 ≤ −2.

Difference Constraints Continued

shortest paths

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properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1


diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


The triangle inequality for shortest paths is adifference constraint. δ(s, v) ≤ δ(s, u) +w(u, v) isequivalent to δ(s, v)− δ(s, u) ≤ w(u, v).

A set of difference constraints xj − xi ≤ bk can bereduced to a weighted graph by w(vi, vj) = bk andw(s, vj) = w(s, vi) = 0.

There is a solution for single-source shortest pathsif and only if there is a solution to the differenceconstraints.

Difference Constraints Example

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2


all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


x1 − x2 ≤ 0x1 − x5 ≤ −1x2 − x5 ≤ 1x3 − x1 ≤ 5x4 − x1 ≤ 4x4 − x3 ≤ −1x5 − x3 ≤ −3x5 − x4 ≤ −3

All-Pairs Shortest Paths

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

⊲ all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


The all-pairs shortest-paths problem is finding theshortest path for each pair of vertices. Algorithmsfor this problem can be adapted for transitiveclosure.

Let δ(i, j,m) be the shortest length from i to j

using ≤ m edges.

δ(i, j, 1) =

0 if i = j

w(i, j) if (i, j) is an edge∞ otherwise

δ(i, j, 2m) =n

mink=1

(δ(i, k,m) + δ(k, j,m))

All-Pairs Recursion

shortest paths

example graphs

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properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

⊲ all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


This recursive definition is correct.

Basis: Equation for δ(i, j, 1) is correct.Assume: δ(i, j,m) is correct.Show: Equation for δ(i, j, 2m) is correct.Induction: Optimal path of ≤ 2m edges consists

of two optimal paths of ≤ m edges.

This leads to a O(V 3 lg V ) algorithm.

All-Pairs Algorithm

shortest paths

example graphs

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properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

⊲ all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


All-Pairs(G,w)n ← |G.V |D ← an n× n matrix initialized to δ(i, j, 1)m ← 1while m < n− 1

Extend(D,n)m ← 2m

return D

Extend(D,n)for i ← 1 to n

for j ← 1 to n

for k ← 1 to n

D[i, j] ← min(D[i, j], D[i, k] +D[k, j])

All-Pairs Example

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

⊲ all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Floyd-Warshall Algorithm

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

⊲ floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Floyd-Warshall(G,w)n ← |G.V |D ← an n× n matrix initialized to δ(i, j, 1)for k ← 1 to n

for i ← 1 to n

for j ← 1 to n

D[i, j] ← min(D[i, j], D[i, k] +D[k, j])return D

Floyd-Warshall is O(V 3).

For transitive closure, use a 0/1 matrix and:D[i, j] ← D[i, j] ∨ (D[i, k] ∧D[k, j])

Floyd-Warshall Analysis

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1


floyd-warshall 3

johnson 1

johnson 2

johnson 3

johnson 4


Let λ(i, j, k) be the shortest distance from i to j

with all intermediate vertices ≤ k.

Basis: D[i, j] ≤ λ(i, j, 0) due to initialization.Assume: D[i, j] ≤ λ(i, j, k−1) before kth iteration.Show: D[i, j] ≤ λ(i, j, k) after kth iteration.Induction: Either λ(i, j, k) = λ(i, j, k − 1) or

λ(i, j, k) = λ(i, k, k−1) + λ(k, j, k−1)

Floyd-Warshall Example

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2


johnson 1

johnson 2

johnson 3

johnson 4


Johnson’s Algorithm for Sparse Graphs

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

⊲ johnson 1

johnson 2

johnson 3

johnson 4


Idea of Johnson’s Algorithm:Run Dijkstra’s algorithm n times.Obtain O(V E lg V ) time using binary heaps.Better than O(V 3) if E is o(V 2)

Problem: Negative weights.

Solution: Clever way to transform the graph sothat all weights are positive

Johnson’s Algorithm

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

⊲ johnson 2

johnson 3

johnson 4


Johnson(G,w, s)(G′, w′) ← (G,w) with new vertex s, w′(s, v) = 0if Bellman-Ford(G′, w′, s) = false

error negative-weight cyclefor (u, v) ∈ G

w(u, v) ← w(u, v) + δ′(s, u)− δ′(s, v)for each vertex u in G

Dijkstra(G, w, u)for each vertex v in G

D[u, v] ← δ(u, v)− δ′(s, u) + δ′(s, v)return D

Johnson’s Algorithm Analysis

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

⊲ johnson 3

johnson 4


Triangle inequality, δ′(s, v) ≤ δ′(s, u) + w(u, v),implies 0 ≤ δ′(s, u) + w(u, v)− δ′(s, v) = w(u, v).Thus, w(u, v) ≥ 0 for all edges (u, v).

For any path p from u to v,w(p) = w(p) + δ′(s, u)− δ′(s, v) because of thetelescoping sum.

Johnson is O(V E lg V ) if binary heaps are usedfor Dijkstra.

Johnson Example

shortest paths

example graphs

bad graph

properties 1

properties 2

basic procedures 1

basic procedures 2

dijkstra 1

dijkstra 2

bellman-ford 1

bellman-ford 2

diff constraints 1

diff constraints 2

diff constraints 3

all pairs

all pairs 2

all pairs 3

all pairs 4

floyd-warshall 1

floyd-warshall 2

floyd-warshall 3

johnson 1

johnson 2

johnson 3

⊲ johnson 4


shortest paths - university of texas at san antoniobylander/cs5633/notes/shortest-paths.pdfall-pairs...

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