1508

1501-1600/1508_range_sum_of_sorted_subarray_sums.rb

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+# @param {Integer[]} nums
+# @param {Integer} n
+# @param {Integer} left
+# @param {Integer} right
+# @return {Integer}
+def range_sum(nums, n, left, right)
+ sums = []
+ (0...nums.size).each { |i|
+ sum = 0
+ (i...nums.size).each { |j|
+ sum += nums[j]
+ sums << sum
+ }
+ }
+
+ sums.sort!
+
+ mod = 1000000007
+ sum = 0
+ (left-1..right-1).each { |i|
+ sum = (sum + sums[i]) % mod
+ }
+
+ sum
+end
+
+
+# binary-search approach
+# O(nlogn)
+def range_sum(nums, n, left, right)
+ # count on sum-array which computed the sum of all non-empty continuous subarrays from the array
+ count_sub_arrays_have_sum_less_than_or_equal = lambda { |max_sum|
+ count = 0
+ total_sum = 0 # sum of all sum of sub-arrays <= max_sum
+ window_sum = 0 # maintain the window [i..j] that has sum <= max_sum
+ total_window_sum = 0 # sum of all sum of sub-arrays (in window) <= max_sum
+ i = j = 0
+ while j < nums.size
+ # for each new j
+ # ..i.....[j]
+ # [ ] <----- all of them is counted into total_window_sum at previous j
+ # [ ] <----- now with new j, we can calculate total_window_sum
+ # [ ] <----- by adding nums[j] for each of them 
+ # ... <-----
+ total_window_sum += nums[j] * (j-i+1)
+ # extend window
+ window_sum += nums[j]
+ # adjust windown [i..j] so the sum <= max_sum by shift-right i
+ while window_sum > max_sum
+ # each time we shift-right i by 1
+ # ...i......j
+ # x[ ] <--- remove i+1
+ # x[ ] <--- remove i+2
+ # x[ ] <--- remove i+3
+ # ..... <--- and so on ... so we end up removing window_sum 
+ total_window_sum -= window_sum
+ window_sum -= nums[i]
+ i += 1
+ end
+
+ total_sum += total_window_sum
+ count += j-i+1 # for each window [i..j], we have j-i+1 sub-arrays
+ j += 1
+ end
+
+ [count, total_sum]
+ }
+
+ min = nums.min
+ max = nums.sum
+
+ # suppose sum-array is sorted, all values will be in range [min..max]
+ # we can find sum-array[i] by binary-searching in [min..max] which max-value that
+ # the number of {sum-sub-array <= value} is >= i + 1
+ # ......value......
+ # ......i
+ #[i+1 sum]
+ #
+ # sum of sum-array[0..index]
+ find_sum_to_index = lambda { |index| # 0-indexed
+ low = min
+ high = max
+
+ while low < high
+ m = (low + high)/2
+ count, sum = count_sub_arrays_have_sum_less_than_or_equal.call(m)
+ if count < index + 1
+ low = m + 1
+ else
+ high = m
+ end
+ end
+
+ # now low is the max-value we need to find
+ # since we count all sums that <= low, so there're some dup lows
+ # we need to cut them
+ # .....low low low ....
+ # .....index....count
+ # [ ]--------- 
+ count, sum = count_sub_arrays_have_sum_less_than_or_equal.call(low)
+ sum - low * (count - index - 1)
+ }
+
+ # 0-indexed
+ # .... left-2 .... right-1 ....
+ # [ ]
+ # [ |-----------]
+ (find_sum_to_index.call(right-1) - find_sum_to_index.call(left-2)) % 1000000007
+end