Couldn't even get a brute force working, which was pretty embarrassing.
Model the problem as a dynamic programming (DP) problem where dp[i] represents the number of valid combinations summing to amount i. Iterate over denominations in sorted order, and for each denomination, update dp from that denomination up to the target, ensuring that when adding a new denomination, the previous denomination used (if any) differs by at least D. Finally, return dp[target] modulo 1000000007.
Pro tip: Clarify whether the constraint applies to any two distinct denominations in the combination or only to consecutive denominations in sorted order; the latter is more common and simplifies the DP. Also, mention that sorting the denominations and using a 2D DP state (amount, last denomination index) elegantly handles the constraint.
Restate the problem: count distinct combinations of unlimited coins summing to target, with the extra rule that any two distinct denominations used must differ by at least D. Clarify edge cases: D=0 means no restriction, D larger than max difference may yield zero combinations.
Sort the denominations ascending. Define dp[i][j] as the number of valid combinations summing to amount i where the last (largest) denomination used has index j. This captures the constraint because any new denomination k must satisfy denom[k] - denom[j] >= D.
Initialize dp[0][j] = 1 for all j (empty combination) or handle base case separately. For each amount i from 1 to target, and for each denomination j, if i >= denom[j], dp[i][j] = sum over k <= j with denom[j] - denom[k] >= D of dp[i - denom[j]][k] plus dp[i - denom[j]][j] (using same denomination again).
Optimize the transition using prefix sums or by iterating denominations in order and maintaining cumulative sums. The final answer is sum over all j of dp[target][j] modulo 1000000007.
Time complexity O(target * n) with optimization, space O(target * n) or O(target) with careful iteration. Test with small examples, including D=0 and cases where no combination exists.
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