Big O and Algorithms (Part 2)

Discrete Structures (CS 173)Madhsuudan Parthasarathy, University of Illinois 1

by Samuel Monnier http://algorithmic-worlds.net/

Last class• Big-O

– Model algorithm efficiency as a function of the parameter size

– Asymptotic analysis: how does it perform for large inputs?

– Keep only dominant terms, ignoring coefficients

– Applies to runtimes or memory usage or any other resources

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This class• Midterm review: induction form and

strategies

• Analysis of algorithm complexity– For loops– While loops– Recursion

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Algorithms• Practice analyzing runtime based on

pseudocode

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Key concept: instructions in the loops dominate

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Key concept: runtime can be in terms of multiple input parameters

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Key concept: n/2 recursion

http://www.youtube.com/watch?v=XaqR3G_NVoo

Reachability algorithm reachable(G: graph; s,t: nodes in G) if (s=t) return true; Unmark all nodes in G. M=emptylist Mark node s and add it to M while (M is not empty) { p=pop(M) for every node q in neighbor(p) in G if q=t return true; else if q is not marked, mark q and add it to M } return false

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Binary search

Searching a sorted array A[1..n]

binary_search(int A[], int key, int imin, int imax) { if (imax < imin) return -1; else { int imid = midpoint(imin, imax); if (A[imid] > key) return binsearch(A, key, imin, mid-1); else if (A[imid] < key) binsearch(A, key, imid+1, imax); else return imid; }}

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Binary search

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12http://introcs.cs.princeton.edu/java/23recursion/hanoi/

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Multiplying large numbersNaïve divide and conquer:

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Multiplying large numbersMore clever divide and conquer (Karatsuba)

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Thursday

• We won’t do NP next lecture.

• We will do advanced recursion/induction instead

• No reading for Thu.

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