algorithm - search in array with specific properties -

i have 2d array in values monotonic. how find (x,y) |f(x,y) – v1| < t in best way.

searching value in matrix sorted rows , columns classic problem. initialize row maximum value , column minimum value. examine matrix entry @ row, column. if it's less target value, increase column one. if it's greater target value, decrease row one. number of matrix entries inspected @ #rows + #columns - 1.

if search continued after target value found decreasing row 1 (respectively, increasing column one), obtain byproduct a determination of elements less than (respectively, less or equal to) the target value. perform search algorithm 2|v| times different target values (less or equal v_i - t, less v_i + t) solve problem o(|v|n) matrix elements inspected.

various optimizations possible. obvious cache matrix values. is, instead of stepping one, use unbounded binary search determine appropriate increment/decrement. it's unclear in prospect whether higher worst-case constant factor overcome savings resulting large changes in neighboring entries.

example using mooing duck's instance:

to elements in range (48, 52),   elements less or equal 48     , (separately) elements less 52.  looking elements less or equal 48:     go right (>) if current element less or equal 48.     go down (v) if current element greater 48  50   60   70   80   90  100  v 40 > 51   60   70   80   90       v 30   50   52   55   70   80       v 30   40 > 45 > 46 > 51   52                      v 30   40   42   45   50   51                      v 30   40   41   44   49   50                      v                   done (access out of bounds)  elements less or equal 48 in submatrix containing   examined element found less or equal 48     , elements left , below.