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Count the subarrays that sum to k, when values can be negative

Explain why a sliding window fails, then solve it in O(n) with prefix sums and a hash map.

TRT practice prompt — not a verified question from a named employer.

The Runtime Theory Team2 min read
Learn it firstPrefix sumsPro — members onlyStore the running total at every position once, and the sum of any range becomes one subtraction. Combined with a hash map, prefix sums count subarrays with a given sum, even with negative numbers.

A strong answer

A strong candidate frames Count the subarrays that sum to k, when values can be negative around the exact memory layout, compiler semantics, and hardware execution model covered in Prefix sums. Explain why a sliding window fails, then solve it in O(n) with prefix sums and a hash map.

A prefix sum array stores, at each position, the total of everything before it. With P[0] = 0 and P[i + 1] = P[i] + a[i], entry P[i] is the sum of the first i values. Building it takes one pass. After that, the sum of any range is the difference of two entries:

The trace above builds P for six values, then answers two range sums with one subtraction each. The extra leading zero is what makes a range starting at index 0 work without a special case, and it is the most common source of off-by-one bugs: decide once that P[i] means "the first i values" and derive every formula from that.

In a production interview, always connect the specification rule to its concrete runtime consequence: state what lives on the stack versus the heap or static storage segment, how the compiler or operating system validates alignments, lifetimes, and error codes, and what happens under adversarial or boundary inputs (such as zero-length buffers, integer overflow, concurrent mutation, or partial I/O). Backing your explanation with a small trace of register/memory state transitions immediately separates senior systems engineers from candidates who only memorized syntax.

What the interviewer is listening for

  • Mastery of core invariant: Build a prefix-sum array and answer range-sum queries in O(1).
  • Mastery of core invariant: Count subarrays with a target sum using prefix sums and a hash map.
  • Mastery of core invariant: Extend the idea to two dimensions, to products, and to range updates with difference arrays.
  • Hardware and ABI awareness: Explaining how sizes, alignments, calling conventions, or kernel boundaries behave on real 64-bit targets rather than relying on implementation-defined assumptions.
  • Defensive engineering: Naming the exact compiler flags (-Wall -Wextra -Werror), static analyzers, or runtime sanitizers (-fsanitize=address,undefined,thread) that verify correctness in CI.

Common mistakes

  • Explaining only surface syntax without tracing the underlying memory addresses, ownership lifetime, or kernel/user-space state transitions.
  • Assuming undefined behavior (such as signed overflow, uninitialized reads, strict-aliasing violations, or dangling pointers) will "just work" because it passed an unoptimized -O0 debug build.
  • Ignoring return codes, partial reads/writes, alignment padding, or cleanup paths when an allocation or syscall fails midway through a function.

This answer walks

Practice follow-ups

  1. 01Why does the map start with {0: 1}?
  2. 02How would you return the longest such subarray instead of the count?
  3. 03How would you count subarrays whose sum is divisible by m?
  4. 04What changes if the input is a stream and memory is limited?

More interviews in this track

hard · dsaHow would you count distinct users across a fleet of services in constant memory?hard · dsaWhy can't a sparse table answer range sums?hard · dsaWhy is a lock-free stack still hard to get right?

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