Linear Time Selection
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1 Linear Time Selection Given (x,y coordinates of N houses, where should you build road These lecture slides are adapted from CLRS.. Princeton University COS Theory of Algorithms Spring 0 Kevin Wayne Given (x,y coordinates of N houses, where should you build road n n Given (x,y coordinates of N houses, where should you build road n n Decreases total cost by (n -n ε ε
2 Given (x,y coordinates of N houses, where should you build road n n Given (x,y coordinates of N houses, where should you build road Solution: put street at y coordinates. Order Statistics Select Given N linearly ordered elements, find i th smallest element. Min: i = Max: i = N Median: i = (N+ / and (N+ / O(N for min or max. O(N log N comparisons by sorting. O(N log i with heaps. Can we do in worst-case O(N comparisons? Surprisingly, yes. (Blum, Floyd, Pratt, Rivest, Tarjan, Assumption to make presentation cleaner. All items have distinct values. Similar to quicksort, but throw away useless "half" at each iteration. Select i th smallest element from a, a. x (N, a, a,..., a N k rank(x if (i == k return x Select (i th, N, a, a Want to choose x so that x is (roughly the ith largest. else if (i < k b[] all items of a[] less than x return Select(i th, k-, b, b,..., b k- else if (i > k c[] all items of a[] greater than x return Select((i-k th, N-k, c, c,..., c N-k x = partition element
3 (. Divide N elements into N/ groups of elements each, plus extra. (. Divide N elements into N/ groups of elements each, plus extra. Brute force sort each of the -element groups N = Select (. Divide N elements into N/ groups of elements each, plus extra. Brute force sort each of the -element groups. Find x = " " by Select( on N/. Select (i th, N, a, a if (N is small use mergesort Divide a[] into groups of, and let m, m,..., m N/ be list of x Select(N/, m, m,..., m N/ 0 0 k rank(x if (i == k // Case return x 0 else if (i < k // Case b[] all items of a[] less than x return Select(i th, k-, b, b,..., b k- else if (i > k // Case c[] all items of a[] greater than x return Select((i-k th, N-k, c, c,..., c N-k
4 Crux of proof: delete roughly % of elements by partitioning. At least / of element x at least N / / = N / x Crux of proof: delete roughly % of elements by partitioning. At least / of element x at least N / / = N / x At least N / elements x Crux of proof: delete roughly % of elements by partitioning. At least / of element x at least N / / = N / x At least N / elements x. At least N / elements x Crux of proof: delete roughly % of elements by partitioning. At least / of element x at least N / / = N / x At least N / elements x. At least N / elements x. Select( called recursively (Case or with at most N - N / elements. C(N = # comparisons on a file of size N. C ( N C ( N / + C ( N N / + O( N recursive select insertion sort 0 Now, solve recurrence. Apply master theorem? Assume N is a power of? Assume C(N is monotone non-decreasing?
5 Analysis of selection recurrence. T(N = # comparisons on a file of size N. T(N is monotone, but C(N is not! 0cN T( N T( N / + T( N N / + cn Claim: T(N 0cN. Base case: N < 0. Inductive hypothesis: assume true for,,..., N-. Induction step: for N 0, we have: if N < 0 otherwise Linear Time Selection Postmortem Practical considerations. Constant (currently too large to be useful. Practical variant: choose random partition element. O(N expected running time ala quicksort. Open problem: guaranteed O(N with better constant. Quicksort. Worst case O(N log N if always partition on median. Justifies practical variants: median-of-, median-of-. T ( N = T ( N / + T ( N N / + cn 0c N / + 0c( N N / + cn 0c( N / + 0c( N 0c( N / + cn 0cN For n 0, N / N /.
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