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Young tableau

·416 words·2 mins
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Definition #

已知一个2维矩阵,其中的元素每一行从左至右依次增加,每一列从上到下依次增加。即对于矩阵Table有Table[i][j] ≤Table[i][j + 1], Table[i][j] ≤ Table[i + 1][j],我们也称这样的矩阵为杨氏矩阵。

Insert #

因为每个元素的下面一个元素和右面一个元素都会比当前元素大,所以右下角的元素是最大的一个元素。所以我们将元素放到矩阵的右下角,然后再来调整元素的位置。我们将这个元素与它上面的元素和左面的元素进行比较,将最大的那个元素与这个元素进行交换,如果这个元素就是最大的话,则已经插入正确的位置。若上面和左面的两个元素相等且大于这个元素,则我们可以交换哪一个都可以。

Delete #

我们要把矩阵中指定的一个元素去掉,那么我们需要调整它右面和下面的元素来符合杨氏矩阵的特性。所以我们不妨将要删除的元素置为NAN。我们将这个元素与他右面和下面的元素中最小的那个进行交换(类似insert操作)

Find #

我们从右上角开始来查找,目的元素比当前元素大则向下查,比当前元素小则向左查。这样们可以在2*n的次数内找到想要的元素。

Modify #

我们将这个重新赋值的元素和它四周的元素进行比较,通过交换调整位置来满足杨氏矩阵的特性。

#-*-coding:utf-8-*-
from numpy import *

def insert(m, value, i, j):
	m[i][j] = value
	largesti = i
	largestj = j

	if i-1>=0 and (isnan(m[i-1][j]) or m[i-1][j] > m[i][j]):
		largesti = i-1
		largestj = j
	if j-1>=0 and (isnan(m[i][j-1]) or m[i][j-1] > m[largesti][largestj]):
		largesti = i
		largestj = j - 1
	if i!=largesti or j!=largestj:
		temp = m[i][j]
		m[i][j] = m[largesti][largestj]
		m[largesti][largestj] = m[i][j]
		insert(m,value,largesti,largestj)

def delete(m, i, j):
	m[i][j] = NAN
	mini = i
	minj = j
	if i+1<m.shape[0]:
		mini = i+1
		minj = j
	if j+1<m.shape[1] and (isnan(m[mini][minj]) or m[i][j+1] < m[mini][minj]):
		mini = i
		minj = j+1

	if mini!=i or minj!=j:
		temp = m[i][j]
		m[i][j] = m[mini][minj]
		m[mini][minj] = temp
		delete(m, mini, minj)

def find(m,value):
	i = 0
	j = m.shape[1] - 1
	while i<m.shape[0] and j >=0:
		if isnan(m[i][j]) or m[i][j] > value:
			j = j -1
		elif isnan(value) or m[i][j] < value:
			i = i+ 1
		else:
			return True
	return False

def modify(m, i, j, value):
	m[i][j] = value
	nexti = i
	nextj = j
	if i-1>=0 and m[i-1][j] > m[i][j]:
		nexti = i-1
		nextj = j
	if j-1>=0 and m[i][j-1] > m[nexti][nextj]:
		nexti = i
		nextj = j-1

	if i+1<m.shape[0] and m[i][j] > m[i+1][j]:
		nexti = i+1
		nextj = j
	if j+1<m.shape[1] and( isnan(m[nexti][nextj]) or m[i][j+1] < m[nexti][nextj]):
		nexti = i
		nextj = j+1

	if nexti!=i or nextj!=j:
		temp = m[i][j]
		m[i][j] = m[nexti][nextj]
		m[nexti][nextj] = temp
		modify(m, nexti, nextj, value)

if __name__ == '__main__':
	m = array([[2,4,6,NAN],[3,7,10,NAN],[5,12,NAN,NAN],[8,NAN,NAN,NAN]])
	h,v = m.shape
	print 'matrix'
	print m
	print '-'*24
	print 'after insert 7'
	insert(m,7,h-1,v-1)
	print m
	print '-'*24
	print 'after delete m[0][0]'
	delete(m,0,0)
	print m
	print '-'*24
	print 'if 12 in the matrix'
	print find(m,12)
	print '-'*24
	print 'after update m[1][1] to 1'
	modify(m, 1, 1,1)
	print m

脚本输出

matrix
[[  2.   4.   6.  nan]
 [  3.   7.  10.  nan]
 [  5.  12.  nan  nan]
 [  8.  nan  nan  nan]]
------------------------
after insert 7
[[  2.   4.   6.  nan]
 [  3.   7.  10.  nan]
 [  5.   7.  nan  nan]
 [  8.  12.  nan  nan]]
------------------------
after delete m[0][0]
[[  3.   4.   6.  nan]
 [  5.   7.  10.  nan]
 [  7.  12.  nan  nan]
 [  8.  nan  nan  nan]]
------------------------
if 12 in the matrix
True
------------------------
after update m[1][1] to 1
[[  1.   4.   6.  nan]
 [  3.   5.  10.  nan]
 [  7.  12.  nan  nan]
 [  8.  nan  nan  nan]]