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Contest - PKU Local 2007 (POJ Monthly--2007.04.28)
Start time: 2007-04-28 12:00:00 End time: 2007-04-28 17:00:00
Current System Time: 2022-05-24 14:43:14.753 Contest Status: Ended
|3218 Problem A||Alignments|
|3219 Problem B||Binomial Coefficients|
|3220 Problem C||Context-Free Languages|
|3221 Problem D||Diamond Puzzle|
|3222 Problem E||Edge Pairing|
|3223 Problem F||Football Game|
|3224 Problem G||Go for Lab Cup!|
|3225 Problem H||Help with Intervals|
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Problem A: Alignments
The four alignments have to be handled separately. Possibly the most tricky point lies in the calculation of gap widths, which demands some care.
Problem B: Binomial Coefficients
The parity of C(n, k) can be determined by calculating the exponent of 2 in its factorization.
Problem C: Context-Free Language
An efficient solution is based on dynamic programming. We first decide what variables each terminal in the string can be derived from using rules of the form A → a. When we have decided information about derivation for all substrings of length not longer than k, we can next decide that for all substrings of length k + 1 by trying to divide each of them into two parts to match the rules of the form A → BC. The string belongs to the language of the grammar if and only if it can be derived from the start symbol S.
Problem D: Diamond Puzzle
The minimum numbers of steps required for every configuration of the puzzle can be precomputed by performing a (backward) breadth-first search from the final configuration and stored in a table. Rest of the job is merely table lookups.
Problem E: Edge Pairing
A simple undirected graph with an even number of edges can always have its edges grouped into incident pairs. Below is a proof by construction.
Let r be some vertex in the graph. We grow a depth-first search tree T rooted at r. Since the graph is connected, T actually spans every vertex. We then construct the pairing bottom up, starting from the leaves. We mark all edges connecting a leave and its parent as unpaired. When working on some vertex x, we count how many unpaired edges connect x and its descendants. If the number of such edges is even, they will all be paired using x as the shared vertex, then the tree edge connecting x and its parent will be marked as unpaired; otherwise pairing fails leaving a single edge, which will be paired with that tree edge. The latter case never occurs with r. The reason is that when we work on r, all edges not connecting r and some of its descendants will have been paired, totaling to an even count. If at this moment pairing leaves a last unpaired edge, this would imply that the graph has an odd number of edges, contradicting the condition that the graph has an even number of edges.
Problem F: Football Game
Let’s first consider the simplified case where P is allowed to be anywhere on the straight line through UV.
In the above figure, a circle through A and B touches UV at tangent point Q, which is to the right of R, the intersection of UV and the extension of BA. We claim that
These three lemmas allow the employment of the bisection method. And the method also works with the original case where the choice of P is restricted to between UV.
Additionally, the tangent point Q can also be analytically determined taking advantage of the fact that Q is the projection of O, the center of the circle, on UV. Since O is equidistant to A, B and UV, it lies on the parabolas with A and B as the foci and UV as the common directrix, and therefore is one of their intersections. Finding O from the formulæ of parabolas requires only solving a univariate quadratic equation.
Problem G: Go for Lab Cup!
Intended as the easiest one, this problem requires only a straightforward implementation of its ideas.
Problem H: Help with Intervals
Efficient representation and manipulation of the set S requires a subtle choice of data structure. Segment tree is one among possible ones. Each node in the tree stores information about membership in S of its corresponding interval, namely whether the interval completely, partially or do not belong to S and whether membership is inverted in this interval (used for quick handling of symmetric difference). Information stored in each node is carefully maintained in each operation. Finally the composition of S can be extracted from the tree.
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