sparsity

# sparsity - OR 3300/5300 Optimization I Professor Bland Fall...

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OR 3300/5300 Optimization I Fall 2011 Professor Bland Sparsity and Packed Storage Assume we have an m × n constraint matrix A , where m and n are large, but σ (the fraction of entries in A that are nonzero) is very small. Then rather than store A explicitly, we can store only the nonzero entries by the following scheme. The arrays V AL ( · ) and ROW ( · ) each have length equal to the number of nonzeros in A , which is σmn . We scan A by columns, starting from column 1 and enter sequentially the value of each nonzero entry and its row number in V AL ( ) and ROW ( ) respectively. Consider an example where: m = 1 , 000; n = 10 , 000, σmn = 125 , 000 and column 1 has 4 nonzero entries: a 72 , 1 = 1; a 416 , 1 = - 1; a 483 , 1 = 6; a 640 , 1 = 3 column 2 has 5 nonzero entries: a 46 , 2 = . 5; a 47 , 2 = 4; a 189 , 2 = - 2; a 352 , 2 = 1; a 810 , 2 = 1 column 3 has 8 nonzero entries: a 212 , 3 = 2 , . . . . Then the arrays V AL ( ) and ROW ( ) begin with i: 1 2 3 4 5 6 7 8 9 10 VAL(i): 1 -1 6 3 .5 4 -2 1 1 2 · · · ROW(i): 72 416 483 640 46 47 189 352 810 212 · · · These arrays each have length σmn = 125 , 000. Every entry in ROW ( ) is an integer

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