Interval DP Introduction

Interval dynamic programming (DP) problems are a unique subset of DP problems that are about linear sequences, like arrays or strings.

While many DP problems have states based on prefixes, interval DP has states on every possible subarray within the given sequence.

In interval DP problems, we'll need to solve the problem on every subarray (interval), hence the name.

Introducing Example Problem: Coin Game

Consider this: You and a friend are seated before n coins lined up in a straight row. Each coin has a unique value, coins[i]. Taking turns, you both can pick a coin from either end of the line, adding its value to your total score. However, once a coin is selected, it's removed from the line. Your friend plays flawlessly, always maximizing their score. The question is, what's your best possible score?

As we delve into the fundamentals of interval DP, we'll reference this example to solidify the concepts.

Identifying Problems Suited for Interval DP

Interval DP is a powerful method for solving a particular subset of dynamic programming problems. To employ it effectively, it's crucial to identify when it's the most appropriate technique.

Recognizing the Potential for Dynamic Programming

Before considering interval DP, ensure the problem is related to dynamic programming. Generally, a problem suited for dynamic programming will fall into one of three categories:

1. Optimization problems: Determining the maximum or minimum solution.
2. Counting problems: Calculating the number of possible solutions.
3. "Is it possible" problems: Determining if a solution exists or if some task is achievable.

In the coin game example, we're asked "what's your best possible score?", hinting at some sort of optimization problem, since we're asked to optimize our score.

Refer to this introduction to dynamic programming for a more comprehensive insight.

Key Indicators for Interval DP

Here are some attributes that might suggest the feasibility of interval DP:

• Linear Sequences: Interval DP solve problems that involve linear sequences like arrays or strings.

  In our Coin Game example, we can notice that the input (coins) is given in an array.

• Time Complexity: One tell-tale sign is the allowable input size, which is explained in this video. If the problem's constraints permit a sequence length up to 3000 or so, it hints at an expected O(n^2) solution (or slower), aligning with the nature of interval DP.

  In addition, we'll observe in the coin game problem, the length of the input coins doesn't exceed 1000.

• Pattern Recognition: As with many problem-solving techniques, there's an inherent pattern to the state transitions and base cases in interval DP problems. If, upon trying to fit a problem into the interval DP structure, the states and transitions don't seem to align naturally, then it's possible that interval DP might not be the best approach.

What are the dp states?

Given that interval DP operates on every possible subarray or interval, there are O(n^2) potential states for a sequence of length n.

An interval DP problem almost always requires a 2-dimensional dp array, with the states often represented as dp[l][r]. Here, l and r define the boundaries of the interval being evaluated.

Example: In our Coin Game example, the state could be represented as dp[l][r], indicating the maximum possible score when considering the coins between positions l and r inclusive. Notice that the only difference between this and the original problem is the original problem considers all the coins while this sub-problem considers only a specific interval of coins, from l to r.

It's important to note that while there are O(n^2) potential states, only about half will be valid since l cannot exceed r for the interval to have positive length. Despite this, the memory complexity remains O(n^2) since around half of the n^2 cells are still used.

Because we compute a solution for every possible interval, the time complexity is also O(n^2), however it could vary, depending on the efficiency of the transition.

Base Cases in Interval DP

The base cases or simple sub-problems for interval DP problems typically involve:

• Empty Sequences: Some problems might define behaviors or return values for empty intervals.

• Single Element Intervals: For most interval DP problems, the simplest non-empty interval is a single element. The behavior or value for this smallest interval often forms the base case upon which larger solutions are built.

Referring to the coin game example, the simplest scenario/base case would be when one coin remains in the line. In this scenario, the current player must take that coin.

Transitioning Between States

Transitions in interval DP involve utilizing solutions from smaller intervals to derive solutions for larger ones. Common strategies you may encounter include:

• Interval Splitting: Divide the current interval into two or more smaller intervals. For instance, in solving a problem on interval l to r (inclusive), you might split it at a point m and combine solutions from intervals l to m (inclusive) and m + 1 to r (inclusive).

• Element Removal: Sometimes, in a problem specifically, you may have to remove elements as some operation. By removing an element in an interval, you'll create smaller intervals and use the solutions from those smaller intervals.

Example: In the Coin Game, when deciding on an interval l to r, the transition could be formulated by considering the result of removing a coin from either the l or r end. By selecting a coin and removing it, you essentially reduce the current interval to a smaller one. The solutions to these smaller intervals, which have already been computed, will then play an important role in determining the optimal decision for the current interval.

Building Intuition through Practice

The key to mastery in any domain is practice. Throughout this chapter, we'll explore examples of interval DP problems. Analyzing these problems will not only enhance your understanding but will also help in building the much-needed intuition to recognize and tackle new interval DP challenges.

By recognizing the signs and familiarizing yourself with the pattern of interval DP problems, you can approach challenges with increased confidence and precision.

Check out the next article for a more comprehensive explanation of Coin Game.