Problem Statement

Originally, the problem is defined in the book as following:

Describe how you could use a single array to implement three stacks. — Cracking the Coding Interview (6th edition)

After some consideration I decided to reshape it. Here is the generalized, yet more specific definition:

Implement a data structure MultiStack that uses a single array to implement arbitrary (N) number of stacks. N is known in advance for every instance of MultiStack.

Feedback I am looking for

Here's the list of things I am interested to hear back (in order of significance):

  1. Design decisions and improvements (as in "better approach(es) performance- and memory-wise").
  2. Code readability.
  3. JavaScript (ES6) language idioms.
  4. Whatever you find important to say that does not fall into three categories mentioned above.

My approach, design, and implementation description

Array-based single stack

Array-based implementations of stacks are very well known. A single stack is normally implemented as an array and an extra field that works as a "pointer to/index of the stack header". One of the array boundaries then works as a bottom of the stack. The other boundary of the array represents the end of available stack "memory".

Here's my sketch for how it works (forgive my handwriting).

single stack

Array-based double-stack

We can also easily use a single array to implement two stacks. Each array boundary represents a bottom of a particular stack (if left boundary is the bottom of the first stack, the right boundary has to be the bottom of the second stack). The directions of growth for the two stacks will be opposite to each other. The end of the available array memory is detected when the two array headers are pointing to adjacent indices.

Here's the illustration:


Array-based N-stack

When we want to implement three or more stacks using a single array things become trickier. This is because we don't have a thing which will work as a clear bottom for the 3rd, 4th, and all the remaining stacks.

Even memory allocation scheme

Of course, we could "allocate" arrays space to each of the stacks simply by evenly distributing the space segments among them. Say, if we have 4 stacks and an array of length 100, we could say: "Segment from 0 to 24 is dedicated to array #1; from 25 to 49 — to array #2; from 50 to 74 — to array #3; and the remaining 75 to 99 — to array #4". If I am not mistaken, this approach is taken in this question, which is about exactly three stacks via a single array.

This certainly would work, but this is only a great solution for the case where all the stacks are populated evenly. However, if one of the stacks is very long, and all the others are very short, we can easily run out of memory for the first array. In general case, not knowing anything about the data, we should not assume this is a great strategy.

Non-even memory allocation scheme

NB: Below I'm using virtual index in a sense of a "pointer" to a data slot/memory slot or a location in the virtual space/stack data (which are the same thing). Sorry for the inconsistent wording here.

Instead of that I decided to use a more dynamic, queue-based allocation technique.

The 0th element of the array is reserved for a virtual index-pointer to the next available memory slot. Next push() to any stack will result in data insertion to the pointed virtual index.

The next N elements of the array are reserved for a virtual index-pointer per stack. These are basically saying where the certain stack's head is.

The rest m (m === data.length - N - 1) array indices are used as a virtual space. The very first index of this section will have a "physical" value of 1+N, and a "virtual" value of 0. The next virtual index (1) has a physical value of 1+N + 2, and so forth. This is because a single virtual index consists of two physical indices. The first value is dedicated to a value hold by the stack entry; and the second value is a virtual index-pointer to the next element in the stack.

The illustration below should help understand what I am trying to say (in case the description is too vague):


A few more notes about this particular implementation.

  • By default, the nextFreeVirtualIndex is pointing to the 0, which is the very first available memory slot.
  • The stack heads are initialized to -1 which is an in indicator of stack being empty.
  • On each push() the code has to properly pre-set the insertion position for the next push() operation. If the current insertion point's references any index, that index will be used (i.e. we're reusing a slot which was used and released at least once in past). If the current insertion point's reference is undefined, the code should use insertionVirtualIndex + 1 which is the first memory slot that has never been used yet.
  • isFull() is not applicable to a particular stack. Instead, a shared outOfMemory() can be used to determine whether we can push to any stack.

Pros & cons


  • truly dynamic allocation of available memory space among the arbitrary number of arrays;
  • therefore, a better fit for the case where a little is known about the stack data in advance;
  • push() and pop()'s complexity have not changed. Both are still O(1).


  • complex implementation, which makes the maintenance harder and potentially leads to bugs (especially if not tested properly);
  • only a half of the memory space actually holds the stack data itself; the other half is "wasted" (spent on the queues/stacks' pointers, which is basically no-data).



class MultiStack {

  get nextFreeVirtualIndex() { return this.data[0]; }
  set nextFreeVirtualIndex(value) { this.data[0] = value; }

  getHeadVirtualIndex(stackNumber) { return this.data[stackNumber + RESERVED_COUNT - 1]; }
  setHeadVirtualIndex(stackNumber, virtualIndex) { return this.data[stackNumber + RESERVED_COUNT - 1] = virtualIndex; }

  getVirtual(virtualIndex) {
    const index = RESERVED_COUNT + this.numberOfStacks + virtualIndex * LENGTH_OF_VIRTUAL;
    return this.data.slice(index, index + LENGTH_OF_VIRTUAL + 1);
  setVirtual(virtualIndex, values) {
    const index = RESERVED_COUNT + this.numberOfStacks + virtualIndex * LENGTH_OF_VIRTUAL;
    this.data.splice(index, LENGTH_OF_VIRTUAL, ...values);

  ) {
    this.numberOfStacks = numberOfStacks;
    this.capacity = capacity || DEFAULT_STACK_CAPACITY;

    this.data = new Array(RESERVED_COUNT + this.numberOfStacks + this.capacity * LENGTH_OF_VIRTUAL);
    this.nextFreeVirtualIndex = 0;
    for (let stackNumber = 1; stackNumber <= this.numberOfStacks; stackNumber++) {
      this.setHeadVirtualIndex(stackNumber, -1);

  push(stackNumber, value) {
    if (this.outOfSpace) {
      throw new Error(`No more space available`);

    const insertionVirtualIndex = this.nextFreeVirtualIndex;
    const [ _, afterNextFreeVirtualIndex ] = this.getVirtual(insertionVirtualIndex);
    const newNextFreeVirtualIndex = afterNextFreeVirtualIndex === undefined ? insertionVirtualIndex + 1 : afterNextFreeVirtualIndex;

    const currentHeadVirtualIndex = this.getHeadVirtualIndex(stackNumber);
    this.setVirtual(insertionVirtualIndex, [ value, currentHeadVirtualIndex ]);
    this.setHeadVirtualIndex(stackNumber, insertionVirtualIndex);

    this.nextFreeVirtualIndex = newNextFreeVirtualIndex;

  pop(stackNumber) {
    if (this.isEmpty(stackNumber)) {
      throw new Error(`Nothing to pop`);

    const retrievalVirtualIndex = this.getHeadVirtualIndex(stackNumber);
    const [ value, tailVirtualIndex ] = this.getVirtual(retrievalVirtualIndex);
    this.setHeadVirtualIndex(stackNumber, tailVirtualIndex);

    const freeVirtualIndex = this.nextFreeVirtualIndex;
    this.setVirtual(retrievalVirtualIndex, [ null, freeVirtualIndex ]);
    this.nextFreeVirtualIndex = retrievalVirtualIndex;

    return value;

  isEmpty(stackNumber) {
    return this.getHeadVirtualIndex(stackNumber) === -1;

  get outOfSpace() {
    return this.nextFreeVirtualIndex === this.capacity;

  printState(prefix) {


End-to-end test

describe(MultiStack.name, () => {
  it(`Should pass a pre-scripted end-to-end test`, () => {
    const capacity = 6;
    const stack = new MultiStack(3, capacity);
    expect(stack.data).toEqual([0, -1, -1, -1, ...new Array(capacity * 2)]);

    stack.push(1, 10);
    expect(stack.data).toEqual([1, 0, -1, -1, 10, -1, ...new Array((capacity - 1) * 2)]);

    stack.push(2, 20);
    expect(stack.data).toEqual([2, 0, 1, -1, 10, -1, 20, -1, ...new Array((capacity - 2) * 2)]);

    stack.push(3, 30);
    expect(stack.data).toEqual([3, 0, 1, 2, 10, -1, 20, -1, 30, -1, ...new Array((capacity - 3) * 2)]);

    stack.push(1, 11);
    expect(stack.data).toEqual([4, 3, 1, 2, 10, -1, 20, -1, 30, -1, 11, 0, ...new Array((capacity - 4) * 2)]);

    stack.push(2, 21);
    expect(stack.data).toEqual([5, 3, 4, 2, 10, -1, 20, -1, 30, -1, 11, 0, 21, 1, ...new Array((capacity - 5) * 2)]);

    let poppedValue;
    poppedValue = stack.pop(1);
    expect(stack.data).toEqual([3, 0, 4, 2, 10, -1, 20, -1, 30, -1, null, 5, 21, 1, ...new Array((capacity - 5) * 2)]);

    poppedValue = stack.pop(1);
    expect(stack.data).toEqual([0, -1, 4, 2, null, 3, 20, -1, 30, -1, null, 5, 21, 1, ...new Array((capacity - 5) * 2)]);

    poppedValue = stack.pop(2);
    expect(stack.data).toEqual([4, -1, 1, 2, null, 3, 20, -1, 30, -1, null, 5, null, 0, ...new Array((capacity - 5) * 2)]);

    stack.push(3, 31);
    expect(stack.data).toEqual([0, -1, 1, 4, null, 3, 20, -1, 30, -1, null, 5, 31, 2, ...new Array((capacity - 5) * 2)]);

    stack.push(3, 32);
    expect(stack.data).toEqual([3, -1, 1, 0, 32, 4, 20, -1, 30, -1, null, 5, 31, 2, ...new Array((capacity - 5) * 2)]);

    stack.push(3, 33);
    expect(stack.data).toEqual([5, -1, 1, 3, 32, 4, 20, -1, 30, -1, 33, 0, 31, 2, ...new Array((capacity - 5) * 2)]);


    stack.push(3, 34);
    expect(stack.data).toEqual([6, -1, 1, 5, 32, 4, 20, -1, 30, -1, 33, 0, 31, 2, 34, 3, ...new Array((capacity - 6) * 2)]);


    expect(() => stack.push(1, 42)).toThrow(`No more space available`);

    expect(() => stack.pop(1)).toThrow(`Nothing to pop`);

Update 1 | 2018-05-10 | to Sumurai8

Even though it was a TypeScript code originally, I'd like it to be reviewed as ES6 (and I keep correcting the syntactic issues to ensure the question is valid). I also indicated ES6 in both "Feedback" section and in tags.

Your idea with buckets is a brilliant way to balance between the rigid upfront-allocated segments I described in "Even memory allocation scheme" on one hand and the bookkeeping-costly linked lists I am using in my code. Thank you so much for showing that! This is the most valuable part of the feedback!!

The end-to-end test was not meant to be exhaustive. Unit testing can be dome much more thoroughly here. It just was not my goal (rather, to illustrate how the things work).

If I had written the code as TypeScript I still would NOT mark data array as private. With data structure implementations I really want the internals to be directly available for observation from test — I'm very firm, it's fine to have internals open this way for consumption in tests. (I'm in agreement with Mark Seemann's thoughts on structural inspection not necessarily breaking encapsulation).

Nevertheless, in real production code (if at all) I would add IMultiStack interface and use it everywhere to access the MultiStack. This is to prevent the accidental undesirable data access by the consumer.

Sorry @Sumurai8, I'm not accepting cheating here. I agree it's definitely an option in the real world, but as a coding exercise, we are supposed to pretend that the array is pretty much all we have for storing the complete stack data. I haven't made it explicit in the problem statement, which is my fault. Otherwise, your idea is what we probably would do without much thinking. :)

I really hate comments in code big time, but this is the case I totally agree that your confusion is an indicator of code readability issue. Comments would help (at cost of maintenance burden).

Long lines are not a big problem on large monitors, but it is indeed possible to soften the issue at least by using proper line return positions. I don't like unnecessary { and } as well as ( and )...

Thanks for pointing out the style issues.

  • 1
    \$\begingroup\$ Should this be tagged with [typescript]? readonly and private don’t look like JS to me. \$\endgroup\$
    – Gerrit0
    Commented May 10, 2018 at 6:25
  • \$\begingroup\$ @Gerrit0 yes, the very original code I have is written in TypeScript. Remembering that TypeScript questions get around ZERO attention compared to JavaScript, I translated the code into ES6 to the best of my ability, but I missed a few things. (I'm planning on doing that in the future, because CR.SE is not appreciating using simultaneous usage of JavaScript AND TypeScript on a single question). I attempted to correct the code -- got rid of private and readonly. \$\endgroup\$ Commented May 10, 2018 at 11:04
  • \$\begingroup\$ Thanks for the down vote, whoever you are. That's really nice of you to not say a word about what is wrong with my question. \$\endgroup\$ Commented May 10, 2018 at 17:30

1 Answer 1


Code style

Disclaimer: I believe you have actually written Typescript instead of ES6. I do not have experience with Typescript, but the concepts are similar enough to things I know that I think I can comment on some of it.

Overall I think your method names are pretty clear, and you split up your code nicely.

Exposing functions

If I read it correctly you are exposing functions like nextFreeVirtualIndex by not declaring it private. The outside world does only need to know about push and pop, and maybe isEmpty, outOfSpace and printState.

Long lines

Consider making your lines of code shorter by adding a newline after braces or ternary operators. Wrapped lines look odd depending on the editor, and too many wrapped lines can make the code look chaotic. Long lines are also harder to read as the readers eyes loose track of the start of the line. Your longest line here is 133 characters, which is somewhat uncomfortably long for me.

Writing readable code is probably more important than adhering to any line length, but putting newlines after a { and before a } are usually low-effort ways to make lines shorter while making the code slightly easier to follow.


complex implementation, which makes the maintenance harder and potentially leads to bugs (especially if not tested properly);

One of the ways of dealing with that is describing what kind of algorithm you are using in comments. Without your written explanation above your code, it would seem really odd to me to see that getVirtual returns an array... with two elements. A comment would help to know what is actually returned here.


The only real optimisation you can do with the current implementation is precalculating this.startOfData = RESERVED_COUNT + this.numberOfStacks. You may also want to rename RESERVED_COUNT to START_OF_STACK_COUNTERS, even though that will look odd in the initialisation of the Array in your constructor.



Since you have written your test verifying the correctness by checking the internal state, you cannot use your test to verify the correctness of a different implementation. Consider writing your test with only the observable behaviour (namely if you pop all the queues it should give you back all your items).

Test against diminishing storage space

A probable pitfall with the problem you are solving is that if you fill all available space, then pop different stacks and fill all available space again, you can end up with less items than the capacity allows. I noticed that you solved this problem by using a linked list in the opposite direction when you pop, but you probably should write a separate test case that makes sure this property is not lost if the code is rewritten.


You already identified that to fill the array to capacity, you need an aweful lot of memory for bookkeeping.


We can always "cheat" and store objects instead of actual values. It does not use less memory, but your array has half the number of elements it used to have. The real upside of using objects is that you can use descriptive keys rather than positions to store information. This will increase the readability of your code.

Less memory with buckets

You can use buckets to decrease the relative amount of memory required for bookkeeping. A bucket contains multiple values for a single stack. You only need one link per bucket.

Your even memory allocation example is an implementation with a bucket size of 1 / n, where your non-even memory allocation is an example with a bucket size of 1. Your code will become slightly more complicated as pushing and popping can now be inside a single bucket, or cross-bucket, and you are wasting up to bucketSize * (numberOfStacks - 1) amount of capacity.

The following is an example of a bucket size of 4 with 4 stacks:

Sample stacks

The corresponding array

You see that the first "bucket" is filled for stack 1, and the last item of stack 1 is on position "12" (namely 3 * bucketSize + position 0) in bucket 3. Stack 2 has the last item on position 7 (namely 1 * bucketSize + position 3). Bucket 4 is the first free bucket. Stack 3 has no buckets assigned.

  • \$\begingroup\$ Thanks for dedicating your time to a thorough review! I don't necessarily agree with everything you said, but at least a few points are totally valid. And the "buckets" idea is ABSOLUTE GOLD! (I updated the Question to specifically address several sections of your answer). \$\endgroup\$ Commented May 10, 2018 at 15:09
  • \$\begingroup\$ Sigh I really want to up vote this answer a few more times. I'll accept it in a few days unless another great answer gets posted here. \$\endgroup\$ Commented May 10, 2018 at 15:12
  • \$\begingroup\$ I'm accepting your answer since it's high quality and there's nobody else to compete with you. Thanks! \$\endgroup\$ Commented May 18, 2018 at 22:33

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