minimum-operations-to-remove-adjacent-ones-in-matrix.py

# Time:  O(E * sqrt(V)) = O(m * n * sqrt(m * n))
# Space: O(V) = O(m * n)

from functools import partial

# Time:  O(E * sqrt(V))
# Space: O(V)
# Source code from http://code.activestate.com/recipes/123641-hopcroft-karp-bipartite-matching/
# Hopcroft-Karp bipartite max-cardinality matching and max independent set
# David Eppstein, UC Irvine, 27 Apr 2002
def bipartiteMatch(graph):
    '''Find maximum cardinality matching of a bipartite graph (U,V,E).
    The input format is a dictionary mapping members of U to a list
    of their neighbors in V.  The output is a triple (M,A,B) where M is a
    dictionary mapping members of V to their matches in U, A is the part
    of the maximum independent set in U, and B is the part of the MIS in V.
    The same object may occur in both U and V, and is treated as two
    distinct vertices if this happens.'''
    
    # initialize greedy matching (redundant, but faster than full search)
    matching = {}
    for u in graph:
        for v in graph[u]:
            if v not in matching:
                matching[v] = u
                break
    
    while 1:
        # structure residual graph into layers
        # pred[u] gives the neighbor in the previous layer for u in U
        # preds[v] gives a list of neighbors in the previous layer for v in V
        # unmatched gives a list of unmatched vertices in final layer of V,
        # and is also used as a flag value for pred[u] when u is in the first layer
        preds = {}
        unmatched = []
        pred = dict([(u,unmatched) for u in graph])
        for v in matching:
            del pred[matching[v]]
        layer = list(pred)
        
        # repeatedly extend layering structure by another pair of layers
        while layer and not unmatched:
            newLayer = {}
            for u in layer:
                for v in graph[u]:
                    if v not in preds:
                        newLayer.setdefault(v,[]).append(u)
            layer = []
            for v in newLayer:
                preds[v] = newLayer[v]
                if v in matching:
                    layer.append(matching[v])
                    pred[matching[v]] = v
                else:
                    unmatched.append(v)
        
        # did we finish layering without finding any alternating paths?
        if not unmatched:
            unlayered = {}
            for u in graph:
                for v in graph[u]:
                    if v not in preds:
                        unlayered[v] = None
            return (matching,list(pred),list(unlayered))

        # recursively search backward through layers to find alternating paths
        # recursion returns true if found path, false otherwise
        def recurse(v):
            if v in preds:
                L = preds[v]
                del preds[v]
                for u in L:
                    if u in pred:
                        pu = pred[u]
                        del pred[u]
                        if pu is unmatched or recurse(pu):
                            matching[v] = u
                            return 1
            return 0
        
        def recurse_iter(v):
            def divide(v):
                if v not in preds:
                    return
                L = preds[v]
                del preds[v]
                for u in L :
                    if u in pred and pred[u] is unmatched:  # early return
                        del pred[u]
                        matching[v] = u
                        ret[0] = True
                        return
                stk.append(partial(conquer, v, iter(L)))

            def conquer(v, it):
                for u in it:
                    if u not in pred:
                        continue
                    pu = pred[u]
                    del pred[u]
                    stk.append(partial(postprocess, v, u, it))
                    stk.append(partial(divide, pu))
                    return

            def postprocess(v, u, it):
                if not ret[0]:
                    stk.append(partial(conquer, v, it))
                    return
                matching[v] = u

            ret, stk = [False], []
            stk.append(partial(divide, v))
            while stk:
                stk.pop()()
            return ret[0]

        for v in unmatched: recurse_iter(v)


import collections


class Solution(object):
    def minimumOperations(self, grid):
        """
        :type grid: List[List[int]]
        :rtype: int
        """
        directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]
        def iter_dfs(grid, i, j, lookup, adj):
            if lookup[i][j]:
                return
            lookup[i][j] = True
            stk = [(i, j, (i+j)%2)]
            while stk:
                i, j, color = stk.pop()
                for di, dj in directions:
                    ni, nj = i+di, j+dj
                    if not (0 <= ni < len(grid) and 0 <= nj < len(grid[0]) and grid[ni][nj]):
                        continue
                    if not color:
                        adj[len(grid[0])*ni+nj].append(len(grid[0])*i+j)
                    if lookup[ni][nj]:
                        continue
                    lookup[ni][nj] = True
                    stk.append((ni, nj, color^1))

        adj = collections.defaultdict(list)
        lookup = [[False]*len(grid[0]) for _ in xrange(len(grid))]
        for i in xrange(len(grid)):
            for j in xrange(len(grid[0])):
                if not grid[i][j]:
                    continue
                iter_dfs(grid, i, j, lookup, adj)
        return len(bipartiteMatch(adj)[0])