[理学]算法导论 第九章 中值和顺序统计.pptVIP

Medians and order statistics （中值和顺序统计） 主要内容 Minimum and maximum最大值/最小值 Order Statistics: The i th order statistic of a set of n elements is the i th smallest element Medians The “halfway point” of the set. We use “the median” to refer to the lower median. Select the ith smallest of n elements (the element with rank i) Minimum: i = 1 Maximum: i = n Median: How fast can we solve the problem? O(n) for min or max O(n lg n) by sorting for general i O(i lgn ) with heaps Next we will see how to do it in O(n) time Simultaneous minimum and maximum Find both the minimum and maximum in some applications同时找到最大值和最小值 Simply find the minimum and maximum independently, using (n-1) ×2 comparisons In fact, at most Process elements in pairs, Comparing pairs of elements first with each other Comparing the smaller to the current minimum and the larger to the current maximum Simultaneous minimum and maximum Selection in expected linear time--Randomized algorithm for finding the i th element Divide and conquer approach Main idea: PARTITION If i k, recurse on the left If i k, recurse on the right Otherwise, output x Select the i = 7th smallest: Analysis of randomized selection Analysis of randomized selection Analysis of randomized selection Analysis of randomized selection Analysis of randomized selection Analysis of randomized selection Summary of randomized selection Selection in worst-case linear time Analysis of selection in worst-case linear time The n elements are represented by small circles,and each group occupies a column The medians of the groups are whitened, and the median-of-medians x is labeled Arrows are drawn from larger elements to smaller Analysis of selection in worst-case linear time Analysis of selection in worst-case linear time Summary of selection in linear time Sorting requires Ω(nlgn) time in the comparison model, even on average, and the linear time sorting algorithms in Chapter make assumptions about the input But the linear time selection algo