The hallmark of the Cooley-Tukey algorithm for Fast Fourier Transform is the butterfly network, which helps reduce O(N^2) computations to O(NlogN). Butterflies are very special graphs entangled in routing [Arora], switching [Chung], shuffling [Yang], and mixing [Czumaj].
Given a weighted connected component, what is the weight of of the maximum independent set, and how many different sets have this weight? A Petersen graph GP(5, 2) with zero vertex weights has three maximal subsets by size, 0 maximum set weight but 76 distinct independent sets with this weight. Continue reading
Gus wants to open franchises of his restaurant, Los Pollos Hermanos, along Central Avenue. There are n possible locations for franchises, where location i is at mile i on Central. Each location i > 1, is thus a distance of 1 mile from the previous one. There are two rules.
- At each location, there can be at most one restaurant, and the profit of a restaurant at location i is p_i.
- Any two restaurants must be at least 2 miles apart.
Min-max heaps were introduced in [ASSS86] as an efficient way to support heap operations for both minimum and maximum values. Structurally, the min-max heap levels alternate between min-heap condition and max-heap, and hence evaluates grandchildren/grandparents during insertion or search. Min-max heaps can also be generalized to find the k-th smallest element in O(1) time.
A problem I was asked: suppose a file that stored many valid IPv4 addresses was corrupted such that all dot in the addresses were removed. Given such a corrupt IP, infer all possible corresponding IP addresses. For example, given 12345, all possible addresses are: 18.104.22.168, 22.214.171.124, 126.96.36.199 and 188.8.131.52. The hint that recursion should be used made the problem easier:
A quick overview of several important comparison-based sorting algorithms:
|Sorting technique||Time (avg)||Time (worst)||Memory|