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Running times continued

Explore the running times of merge sort and other divide-and-conquer algorithms, including recurrence analysis and the Master Theorem. Also, learn about counting inversions and finding the closest pair of points.

bradleytoney
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Running times continued

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  1. Running times continued • some running times are more difficult to analyze • e.g., recursive algorithms • Example: MergeSort

  2. Running times continued Merging two sorted lists Input: Output: Two arrays A = {a1, a2, …, am}, B = {b1, b2, …, bn}, in increasing order Array C containing A [ B, in increasing order • MERGE(A,B) • i=1; j=1; k=1; • am+1=1; bn+1=1; • while (k · m+n) do • if (ai < bj) then • ck=ai; i++; • else • ck=bj; j++; • k++; • RETURN C={c1, c2, …, cm+n} Running time ?

  3. Running times continued Sorting Input: Output: An array X = {x1, x2, …, xn} X sorted in increasing order MergeSort – a divide-and-conquer algorithm • MERGESORT(X,n) • if (n == 1) RETURN X • middle = n/2 (round down) • A = {x1, x2, …, xmiddle} • B = {xmiddle+1, xmiddle+2, …, xn} • As = MERGESORT(A,middle) • Bs = MERGESORT(B,n-middle) • RETURN MERGE(As,Bs)

  4. A recurrence Running time of MergeSort: T(n) How to bound T(n) ? -> “unrolling the recurrence”

  5. A recurrence Running time of MergeSort: T(n) How to bound T(n) ? -> “substitution / induction”

  6. Master Theorem • Let a ¸ 1 and b>1 be constants, f(n) be a function and for positive integers we have a recurrence for T of the form • T(n) = a T(n/b) + f(n), • where n/b is rounded either way. • Then, • If f(n) = O(nlog a/log b - e) for some constant e> 0, then • T(n) = (nlog a/log b). • If f(n) = (nlog a/log b), then T(n) = (nlog a/log blog n). • If f(n) = (nlog a/log b + e) for some constant e > 0, and if af(n/b) · cf(n) for some constant c < 1 (and all sufficiently large n), then • T(n) = (f(n)).

  7. Master Theorem • Let a ¸ 1 and b>1 be constants, f(n) be a function and for positive integers we have a recurrence for T of the form • T(n) = a T(n/b) + f(n), • where n/b is rounded either way. • Then, • If f(n) = O(nlog a/log b - e) for some constant e> 0, then • T(n) = (nlog a/log b). • If f(n) = (nlog a/log b), then T(n) = (nlog a/log blog n). • If f(n) = (nlog a/log b + e) for some constant e > 0, and if af(n/b) · cf(n) for some constant c < 1 (and all sufficiently large n), then • T(n) = (f(n)).

  8. Master Theorem • Let a ¸ 1 and b>1 be constants, f(n) be a function and for positive integers we have a recurrence for T of the form • T(n) = a T(n/b) + f(n), • where n/b is rounded either way. • Then, • If f(n) = O(nlog a/log b - e) for some constant e> 0, then • T(n) = (nlog a/log b). • If f(n) = (nlog a/log b), then T(n) = (nlog a/log blog n). • If f(n) = (nlog a/log b + e) for some constant e > 0, and if af(n/b) · cf(n) for some constant c < 1 (and all sufficiently large n), then • T(n) = (f(n)).

  9. More on sorting Other O(n log n) sorts ? Can do better than O(n log n) ?

  10. A lower-bound on sorting: (n log n) Every comparison-based sort needs at least (n log n) comparisons, thus its running time is (n log n).

  11. Divide-and-conquer algorithms Counting inversions Input: A permutation a1,a2,…,an of 1,2,…,n Output: # inversions, i.e. (i,j) pairs where i<j but ai>aj

  12. Divide-and-conquer algorithms Counting inversions Input: A permutation a1,a2,…,an of 1,2,…,n Output: # inversions, i.e. (i,j) pairs where i<j but ai>aj

  13. Divide-and-conquer algorithms Counting inversions Input: A permutation a1,a2,…,an of 1,2,…,n Output: # inversions, i.e. (i,j) pairs where i<j but ai>aj

  14. Divide-and-conquer algorithms Closest Pair of Points Input: n points in the plane (x1,y1),…,(xn,yn) Output: distance of the closest pair of points, i.e., mini,j (i j) distance of (xi,yi) and (xj,yj)

  15. Divide-and-conquer algorithms Closest Pair of Points Input: n points in the plane (x1,y1),…,(xn,yn) Output: distance of the closest pair of points, i.e., mini,j (i j) distance of (xi,yi) and (xj,yj)

  16. Divide-and-conquer algorithms Closest Pair of Points Input: n points in the plane (x1,y1),…,(xn,yn) Output: distance of the closest pair of points, i.e., mini,j (i j) distance of (xi,yi) and (xj,yj)

  17. Divide-and-conquer algorithms Closest Pair of Points Input: n points in the plane (x1,y1),…,(xn,yn) Output: distance of the closest pair of points, i.e., mini,j (i j) distance of (xi,yi) and (xj,yj)

  18. Divide-and-conquer algorithms Closest Pair of Points Input: n points in the plane (x1,y1),…,(xn,yn) Output: distance of the closest pair of points, i.e., mini,j (i j) distance of (xi,yi) and (xj,yj)

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