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Problem solution · Python

CCC 2015 S3 - Gates

CCC 2015 S3 - Gates: a Python solution using disjoint set union. Learn the idea, check the complexity, and read the full code, with credit to CCCSolutions.

Technique
Disjoint set union
Source
CCCSolutions
Length
80 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Disjoint set union

For CCC 2015 S3 - Gates, the implementation maintains connected components and merges them as relationships are processed.

  1. Give each element a component representative.
  2. Merge representatives when a connection is accepted.
  3. Answer connectivity or component queries from the compressed representatives.

Code notes

  • 80 lines of Python from the credited upstream file ccc15s3.py.
  • The implementation keeps its working state in language-native values and containers.
  • No explicit loop blocks detected.

Complexity

Account for every find and union operation; with path compression and ranked merging, the amortized cost is nearly constant per operation.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from CCCSolutions by CCCSolutions contributors · Milliken Mills High School and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeCCC 2015 S3 - Gates · PythonPython
Use this to learn the idea, then write your own version.
# CCC 2015 Senior 3: Gates## This solution is by Weiwei Zhong## This uses a disjoint-set Data Structure## see  http://community.topcoder.com/tc?module=Static&d1=tutorials&d2=disjointDataStructure# for a good exlaination of the data structure## to help explain this, look at the following input data set:#   6 7 6 1 4 4 6 6 3   (6 gates and 7 planes.)## root is initialized to [0, 0, 0, 0, 0, 0, 0]  (there are more 0's, but forget them)## After reading the number of gates, root is [0, 1, 2, 3, 4, 5, 6]# think of each number being its own set. ## The first plane goes in gate 6 and its found in its own set, so "take it",# by merging it with the set below it. # so set 6 and 5 are merged and root becomes [0, 1, 2, 3, 4, 5, 5]# The way to "read" this is that set 6 doesn't exist any more and if part of the "5" set.# a set is identified with the lowest number in the set.## The same thing happens with the 1 and the 4, they go in their respective gates# and those gates are merged with the sets below them.# this makes root                            [0, 0, 2, 3, 3, 5, 5]## The following 4 finds that 4 is part of the 3 set. We haven't had that before.# In this case merge changes root to become  [0, 0, 2, 2, 3, 5, 5]# (now stop a minute and look at that: merge sets root[find(x)] = find(y)#  or in our case                                 root[find(3)] = find[2]#  which means                                    root[3] = 2# A way to "read" this is that in root, if x = root[x] then gate x is available.# and now after 4 planes, only root 2 and 5 remain. Or if you prefer the world of sets# we have 3 sets (0,1), (2,3,4) and (5,6)## With the next plane wanting 6 again we can see that gate 5 will be taken and we'll be# down to only two sets (0,1) and (2,3,4,5,6), the only "open" gate will be 2.# After the merge, root becomes              [0, 0, 2, 2, 2, 2, 5]# How did that happen? Well it's complex due to the recursion happening in find, BUT# we can see that there is only one open gate now, 2, and that we have two sets.## With the third palne wanting 6, the last open gate, 2 is taken and root becomes#                                            [0, 0, 0, 2, 2, 2, 2]# which is a single set represented by 0, no gates left.## The next plane request for gate 3 stops the program, and i = 6 is printed.## I realize this is as clear as mud: Go to the website and read! :-)  root = [0 for i in range(100001)] #find the set this index belongs todef find(x):     if x!=root[x]:        root[x] = find(root[x])    return root[x] #merge index x into ydef merge(x,y):     root[find(x)] = find(y) file = open("s3.10.in")    numGates = int(file.readline())numPlanes = int(file.readline())for i in range(1,numGates+1):    root[i] = ii = 0keepgoing = Truewhile i < numPlanes and keepgoing:    gateWanted = int(file.readline())    gateWantedRoot = find(gateWanted)    if gateWantedRoot == 0:        print i        keepgoing = False     merge(gateWantedRoot,gateWantedRoot-1)    i = i + 1if keepgoing:        print numPlanes

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