Optimal Binary Search Trees Explained Simply

Initial Thoughts

In the world of data structures, especially when handling large amounts of data, efficiently searching for information is key. This is where Optimal Binary Search Trees (OBST) come into play. Unlike a regular binary search tree, which might lead to longer search times due to uneven tree shapes, an OBST is designed to minimize the expected search cost, making data retrieval faster and more efficient.

For traders, investors, financial analysts, and students, understanding OBST can be especially useful. It helps explain how algorithms optimize search operations—something that underpins many financial data systems where quick decision-making matters. Imagine having to scan through thousands of stock quotes every second; any time saved by an optimal search can translate into better trading outcomes.

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In this article, we’ll cover:

• What problem OBST aims to solve

• How OBST construction works

• The algorithms behind OBST

• Real-world examples and applications

• Complexity and performance analysis

Getting a grasp on these points will sharpen your insight into how intelligent data structures improve processing speed and accuracy, an advantage in any data-heavy task or financial modeling.

Opening to Optimal Binary Search Trees

Optimal Binary Search Trees (OBSTs) play a big role in fine-tuning how quickly we can find items in a sorted dataset. This is especially useful in fields like finance, where rapidly retrieving data can influence decisions that affect trades or investments.

When dealing with large amounts of data, your standard binary search tree (BST) might not always be the fastest. OBSTs aim to cut down the average time spent searching by arranging nodes based on how likely they are to be accessed. Think of it like organizing files on your desk so you grab the most-used ones without fumbling.

Understanding OBSTs is not just academic—it has practical benefits. For example, in building trading algorithms or financial databases, quicker access to key data means faster analysis and better responsiveness. The concepts here will lead you through what a BST is and why optimizing it matters, setting the stage for deeper discussions about building and using OBSTs efficiently.

What Is a Binary Search Tree

A Binary Search Tree is a structured way of storing data that keeps everything in order. Each node in the tree has a value, and it follows two main rules:

• Values in the left subtree are smaller than the node

• Values in the right subtree are larger than the node

This setup makes searching straightforward because you know where to go next just by comparing values. Imagine looking up a stock ticker symbol in an alphabetical list where you only check sections that could possibly contain the symbol.

For example, if you're searching for the symbol "INFY" in a tree, you’ll move left or right starting from the root node based on whether "INFY" comes before or after the node's value alphabetically. This cuts down the possible search area quickly.

However, BSTs can get unbalanced if data isn't inserted carefully. A tree leaning too far to one side can slow searches down, acting more like a linked list.

Why Optimizing Binary Search Trees Matters

Optimization in BSTs is about making your searches as fast as possible on average, not just in the best case. Consider a financial app where certain stocks are looked up way more often than others. A normal BST treats all stocks equally, but an OBST arranges them so the popular stocks are closer to the root, making those searches quicker.

Without optimization, you might waste extra milliseconds on every search, which adds up in high-frequency trading or live data feeds. In stockbroking, those delays could mean missed opportunities.

An optimized tree also reduces the overall cost of searches. Here, "cost" refers to the number of nodes you have to inspect before finding your target. By minimizing this cost, your algorithms run leaner and react faster.

A well-optimized binary search tree is like having your favorite tools within arm’s reach, not buried deep in the toolbox.

In a nutshell, optimizing BSTs is about improving efficiency and saving time where it counts, especially in environments heavy with real-time data like finance.

Problem Statement for Optimal Binary Search Trees

Understanding the problem statement behind Optimal Binary Search Trees (OBSTs) is key to grasping why they're so valuable in certain data handling situations. At its core, the problem focuses on how we can arrange keys in a binary search tree so that the overall cost of searching for these keys is as low as possible. This matters a lot when you have a large dataset and search queries that happen frequently with varying chances for each key.

Most traditional binary search trees don’t consider how often each item gets searched; they treat every search as if it were equally likely. But in real-world applications — think of traders looking for specific stocks or financial analysts querying particular financial records — some data points get accessed way more than others. Ignoring this fact can lead to slower searches overall, wasting precious time.

Understanding Search Costs in BSTs

Search cost in a binary search tree largely depends on the depth at which a key is found. Each step down the tree adds time, especially when you factor in memory access or computational delays. For example, if a trader’s most sought-after company stock data is buried several levels deep, the delay could be significant across thousands of queries. The cost isn't just about the number of steps; it’s about how often these steps need to be taken.

Search costs fluctuate because each key might have a different probability of being searched. A stockbroker frequently looking up high-cap stocks like Reliance or TCS will experience repeated delays if the tree isn't optimized for those keys. A binary search tree that doesn’t consider these varying probabilities can lead to uneven search times, dragging down overall performance.

Goal of Minimizing Expected Search Time

The main goal for an OBST is to minimize the expected search time, which is a weighted average based on how likely each key is to be accessed. Instead of a one-size-fits-all approach, OBST tailors the tree structure to reduce the average cost, not just the worst-case scenario.

Let’s say you have a set of ten financial indicators, each with different chances to be queried by investors. If the indicators that show market volatility tend to be looked up more frequently, the OBST will try to place these keys closer to the tree’s root. This way, when investors or analysts run their queries, the average wait time per search shrinks significantly.

This approach balances the search paths so that the most commonly accessed keys are the quickest to reach. As a result, systems can provide faster responses, reducing lag in decision-making processes—a neat win for traders and analysts alike.

Efficient searching isn’t just about speed—it's about smart speed. OBSTs tackle this by considering real-world usage patterns to deliver better performance where it counts.

In short, the problem statement for OBSTs revolves around restructuring a BST such that the total weighted search cost is minimized, making search operations swift and practical, particularly in finance-related heavy-load environments where every millisecond counts.

Key Concepts Behind Optimal Binary Search Trees

The backbone of creating an optimal binary search tree (OBST) lies in understanding several key ideas that determine how the tree is structured and functions. These concepts aren’t just academic—they translate directly into how effective your search operations will be, which is critical in fast-paced environments like stock trading platforms or real-time financial analysis systems. At the heart of it, OBSTs try to organize data so that commonly searched items are found quicker, cutting down on unnecessary steps.

Probabilities of Searching Keys

A vital piece in building an OBST is grasping the probabilities of searching keys. Think of this as knowing the popularity of each stock ticker or financial instrument in your database. Instead of treating each search equally, OBST assumes you have a set of probabilities that predicts how often a key might be searched. For instance, in a portfolio database, Apple (AAPL) might be searched 30% of the time, whereas a less popular tech stock might only see 2%.

Using these probabilities​ helps tailor the tree to fit real-world usage rather than a uniform or random pattern.

• High-probability keys should ideally be positioned closer to the root of the tree.

• Low-probability keys can be placed deeper without significantly hurting overall search time.

For example, if you have five keys with these search probabilities: 0.4, 0.25, 0.15, 0.12, and 0.08, an OBST will arrange them so that the 0.4 key requires fewer steps to reach than the 0.08 key. This reflects the actual frequency these keys get queried.

Expected Search Cost and Its Calculation

Once probabilities are set, the next major concept is the expected search cost. This is basically the average number of comparisons you'd expect during a search, factoring in where the key is located and how often it's searched. It’s like budgeting time for different queries—more frequent searches should cost you less time on average.

Mathematically, the expected search cost sums the product of each key’s probability and its depth in the tree. The deeper a key, the more comparisons it needs, so higher-depth keys with high probabilities obviously increase this cost.

Expected Search Cost = ∑ (Probability of key i × Depth of key i in the tree)

Here's a quick example: Supposing three keys with probabilities 0.5, 0.3, and 0.2 are placed at depths 1, 2, and 3 respectively. The expected cost would then be:

plaintext (0.5 × 1) + (0.3 × 2) + (0.2 × 3) = 0.5 + 0.6 + 0.6 = 1.7

Here's what's going on:

• Cost(i, j) is the minimal search cost for keys between i and j.

• Cost(i, k-1) and Cost(k+1, j) are the costs of left and right subtrees when k is the root.

• Sum_Probabilities(i, j) is the total probability sum for keys from i to j, accounting for the added depth at this step.

For example, suppose you have keys A, B, and C with respective probabilities 0.2, 0.5, and 0.3. To compute the cost for keys A to C, you check choosing A, then B, then C as roots and pick whichever gives the smallest total cost based on the formula. This straightforward yet powerful approach avoids brute force.

Remember, the key is balancing the tree so frequent searches hit their targets faster, and dynamic programming finds that sweet spot.

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In practical terms, implementing this recurrence in a bottom-up manner means filling out a cost table starting from single keys and expanding outwards to larger ranges. Auxiliary tables store the roots selected for every subproblem, helping reconstruct the optimal tree once all costs are computed.

By using dynamic programming, OBST construction becomes much more manageable and efficient—a must-know technique if you want to plunge into deeper data structure optimization.

Step-by-Step Construction of an Optimal BST

Building an Optimal Binary Search Tree (OBST) isn't just an academic exercise—it's an essential process that directly impacts how quickly you can search through data. Understanding the construction process helps in appreciating how careful arrangement of nodes reduces the average search time, which is especially important in areas like financial analytics where data retrieval speed can affect real-time decision-making.

The step-by-step build hinges on calculating and using specific tables to determine the best structure. These tables guide us in deciding which node should be the root for every subrange of keys to minimize the expected cost. This method leads to a tree that balances search efforts according to how often certain keys are accessed, making retrieval more efficient.

Computing Cost and Root Tables

The foundation for building an OBST lies in two key tables: the Cost Table and the Root Table. The Cost Table stores the expected search costs for every possible sub-tree, while the Root Table keeps track of the root nodes that lead to those minimal costs.

To compute these, first, we define probabilities for each key—how often each is searched—and also probabilities for unsuccessful searches (known as dummy keys). With these inputs, we calculate accumulated probabilities for subranges of keys, which act as weights in our computations.

Next, for each subrange from key i to key j, we test every key k in that range as a candidate root. The cost of choosing k as the root is the sum of three terms:

• Cost of the left subtree (keys i to k-1)

• Cost of the right subtree (keys k+1 to j)

• Total probability of the keys and dummy keys in this range

By iterating through all candidates, we pick the root that yields the smallest total cost. This value and the chosen root’s index are recorded in the Cost and Root Tables respectively.

This dynamic programming approach ensures no recomputation is wasted, and every decision builds upon previously computed results.

Building the Tree from Computed Data

Once the Cost and Root Tables are ready, the next step is assembling the tree itself. Starting from the entire key range, the Root Table tells us which key should be the root. We then recursively apply the process to the left and right subranges of keys, constructing the left and right subtrees.

For example, if the root for keys 1 to 5 is key 3, we create a node with key 3, then look up the Root Table for the best root between keys 1 to 2 to build the left child, and between keys 4 to 5 to build the right child.

This recursive building continues until all keys become nodes with their appropriate left and right children or until subranges contain no keys, culminating in a fully optimized binary search tree.

Building an OBST this way ensures the tree reflects actual access patterns, thereby improving average search times when compared to a regular BST. For traders or financial analysts, it means quicker data lookups, which can be crucial during fast market moves.

In practice, this method is implemented with careful bookkeeping of tables and recursive tree-building functions in languages like Python or Java. This structured approach helps keep the code maintainable and the tree truly optimal according to the provided probabilities.

In the next sections, we'll look at a concrete example that walks through these computational steps and tree construction to solidify your understanding.

Example Illustration of OBST Construction

Seeing theory move into practice is the best way to grasp how optimal binary search trees work. This section shows how you can apply the OBST algorithm to a specific set of keys and probabilities, turning abstract formulas into a tangible example. Understanding this helps demystify the calculations and makes the concept much clearer.

Sample Key Set and Probabilities

Let's start with a key set that's pretty straightforward but rich enough to illustrate the process: suppose you have four keys: A, B, C, and D. Imagine these keys correspond to stock ticker symbols or financial terms, which might be common for analysts. Now, each key has a probability indicating how often that item gets searched or accessed. For instance:

• Key A: 0.15

• Key B: 0.10

• Key C: 0.05

• Key D: 0.30

Along with these, we also define the probabilities for unsuccessful searches (things not in the tree) between and around these keys to handle misses:

• Dummy keys (denoting misses): q0 = 0.05, q1 = 0.10, q2 = 0.05, q3 = 0.10, q4 = 0.10

These probabilities give us the frequency data necessary to build an OBST that minimizes the expected search cost. Without these, the algorithm couldn't optimize since it optimizes based on probability-weighted cost.

Stepwise Formation of Optimal Tree

Building the OBST follows a series of methodical steps, not unlike piecing together a financial portfolio where each asset's weight influences the overall risk and return:

1. Calculate weight sums: First, sum the probabilities for every possible subtree—this means for every continuous subset of keys. For example, from A to B, A to C, and so on.

2. Fill the cost and root matrices: Using dynamic programming, compute the cost of building a tree from these subsets and identify which key should be the root to minimize the cost.

3. Choose roots for subtrees: For each subset, pick the root that yields the lowest expected cost.

4. Construct the tree: With these roots decided, build up the full tree recursively.

To put it into perspective: when you try A as root for keys A-D, its search cost would be based on how likely you look for A and the structural costs of the remaining keys under it. Trying B, C, or D as root changes these costs, so computing each lets us find the best arrangement.

This stepwise process ensures you don’t rely on guesswork but on systematic calculation. It's like balancing your portfolio after precise risk analysis rather than guess-timating.

Breaking down the example:

• Calculate the initial weights for all key ranges.

• Use these to find minimum costs and best roots for one key, then two keys, steadily building upward.

• Final outcome gives you a root, say D, with left subtree rooted at B and so forth, minimizing the average search time.

By working through this example, you're not just learning how the algorithm functions but gaining insight into optimizing real-world search problems where search frequencies are unequal. This can translate directly into more efficient querying systems in databases or faster lookups in trading software.

Understanding every move in this toy example grounds your intuition and equips you better for implementing OBSTs in complex applications like financial data retrieval, making your programs leaner and faster.

Performance and Complexity Analysis

When we talk about optimal binary search trees (OBST), understanding their performance and complexity is key to seeing how well they serve real-world needs. It's one thing to design a tree that theoretically minimizes search cost, but it's another to ensure it can be built and used efficiently in practice. This section sheds light on the actual computational demands of OBSTs and what that means for developers and users.

Time Complexity of OBST Algorithms

Constructing an OBST might sound straightforward, but the computation behind it can be quite taxing, especially for larger sets of keys. The classical dynamic programming method used to build OBSTs has a time complexity of roughly O(n³), where "n" is the number of keys. This cubic time might seem steep, especially compared to the O(n log n) time complexity seen in balanced trees like AVL or Red-Black trees.

To get a sharper picture, imagine you've got 100 keys. Constructing the OBST here involves about a million operations just from the dynamic programming approach (since 100³ = 1,000,000). This intensive calculation means OBSTs are best suited when you have a manageable number of keys and the search probabilities are well understood and stable.

Some variations and optimizations can cut down this cost, like Knuth's optimization, which can reduce the time complexity closer to O(n²). This makes OBST more practical but still not as fast as balanced BSTs for dynamic or very large data sets.

Space Requirements and Optimization

OBST construction doesn't just eat CPU cycles; it also demands significant memory. The dynamic programming table storing costs, roots, and probabilities grows quickly with the number of keys, usually requiring O(n²) space.

For example, if you're analyzing stock price movements and want to optimize search through 500 keys (say, price points or company symbols), this table must hold 250,000 entries or more. This can eat up considerable RAM depending on your system, and might become a bottleneck in resource-limited environments.

Memory optimization tactics often involve careful data structure choices or even pruning methods to reuse or limit data storage. For instance, if search probabilities are sparse or heavily skewed, you might tweak the algorithm to focus only on likely access patterns, reducing overall memory needs.

Keep in mind: While OBSTs aim to minimize expected search costs, their construction phase can be demanding. Knowing the time and space trade-offs can help in deciding whether they fit your data access scenario.

In the next sections, we'll see where OBSTs shine in application despite these challenges, and when other structures might be more appropriate for dynamic or large-scale data environments.

Practical Uses of Optimal Binary Search Trees

Optimal Binary Search Trees (OBSTs) aren't just theoretical constructs—they find real-world uses where reducing search cost can save both time and resources. This section highlights some specific areas where OBSTs offer practical benefits, emphasizing how their design helps manage data queries efficiently. By minimizing the average search time, OBSTs enable smoother and faster data retrieval in applications where speed directly impacts performance.

Applications in Database Indexing

In database systems, quick access to records is critical. Indexes speed up retrieval by pointing directly to data locations without scanning the entire dataset. OBSTs come into play when certain queries or keys are searched more frequently than others. By organizing keys based on their access probabilities, an OBST minimizes the expected number of comparisons.

For instance, in a stock trading application, some Tickers like "RELIANCE" or "TCS" might be requested more often, so placing these closer to the tree’s root reduces lookup time. Instead of a balanced tree that treats all keys equally, an OBST shapes itself around real-world search patterns. This targeted optimization can significantly boost query speeds in databases managing financial instruments or transaction logs.

OBST in Compiler Design and Autocomplete Systems

OBSTs also find a niche in compiler design and autocomplete features, where search efficiency directly affects user experience and runtime.

Compilers often need to quickly access syntax rules or tokens, some appearing far more frequently. An OBST can organize these tokens such that the most commonly used keywords (like "if", "for", or "return") sit closer to the root, speeding up parsing.

Autocomplete tools, especially in financial software where users might search stock symbols or company names, benefit greatly from OBSTs. The system can prioritize suggestions based on how often certain queries are made and the user's typing habits, ensuring that the most probable results show up earliest in the suggestions list.

Quick Note: In both compilers and autocomplete, OBSTs adjust to search frequencies, making the usage feel snappy without wasting compute power on less frequent tokens or suggestions.

In summary, when data access patterns are skewed, OBSTs tailor the search structure to real usage, offering a practical edge in systems demanding quick lookups and responsiveness.

Limitations and Challenges of OBSTs

Understanding the limitations and challenges of Optimal Binary Search Trees (OBSTs) is essential for anyone looking to implement them effectively. While OBSTs minimize the expected search cost by arranging keys based on access probabilities, their practical use can be complicated by various factors. These complexities often make OBSTs less appealing compared to other search tree variants, especially in dynamic or real-time scenarios.

Changing Access Probabilities and Tree Maintenance

One of the biggest challenges with OBSTs lies in their sensitivity to changes in access probabilities. OBSTs depend heavily on the assumption that the search frequencies (or probabilities) of keys are known and relatively stable. But in many real-world applications, like stock price lookup systems or live market data analysis, the frequency of key accesses can fluctuate unpredictably.

When these probabilities shift, the OBST may no longer be optimal, resulting in slower search times until the tree is rebuilt. However, the rebuilding process itself can be costly in terms of time and computation. For example, consider a trading application that constantly monitors the price of thousands of stocks where some stocks suddenly become highly relevant due to market news. The OBST designed with previous access probabilities might fail to provide efficient searches for such key stocks, requiring frequent tree reconstructions.

Maintaining an OBST dynamically can be resource-heavy, making it less practical for systems where access patterns change frequently.

Comparison with Other Search Tree Variants

While OBSTs excel at minimizing expected search costs given fixed probabilities, they are not always the best choice compared to other binary search trees, especially for dynamic data.

• AVL Trees and Red-Black Trees: Both these balanced trees maintain their balance after each insertion or deletion to ensure worst-case logarithmic time for searches. Unlike OBSTs, they don’t rely on access probabilities. This makes them ideal for systems where data changes often and predictability of search time is important, such as database indexing and real-time trading systems.

• Splay Trees: These are self-adjusting trees that bring frequently accessed elements closer to the root without prior knowledge of access frequencies. They adapt automatically, which can be more practical when actual access patterns are unknown or unpredictable.

• Hashing: While not a tree structure, hashing offers average constant-time lookup and is widely used in situations where exact order isn’t necessary. However, hashing may face collisions and doesn’t support range queries as elegantly as BSTs.

To put it simply, OBSTs are perfect when access probabilities are stable and known in advance, but for rapidly changing environments found in financial markets and trading platforms, balanced trees or self-adjusting trees might offer more consistent performance.

In summary, the challenge with OBSTs is their reliance on static probability data and the costly process to update the tree when these probabilities change. For dynamic or real-time applications, alternative search structures often offer better maintainability and consistent performance despite not guaranteeing minimum expected search cost every single time.

Alternatives to Optimal Binary Search Trees

When dealing with data retrieval and storage, Optimal Binary Search Trees (OBSTs) aren’t the only option on the table. While OBSTs aim to minimize the expected search time based on known access probabilities, they come with their own tradeoffs in terms of dynamic adaptability and implementation complexity. It’s important to recognize alternatives that often serve similar purposes but shine under different conditions. Exploring these helps investors, traders, and finance professionals decide which structure fits their use cases best—especially when performance, modification frequency, or memory constraints come into play.

Balanced BSTs like AVL and Red-Black Trees

Balanced Binary Search Trees such as AVL trees and Red-Black trees are widely used alternatives to OBSTs. Unlike OBSTs, these trees don’t rely on prior knowledge of access probabilities. Instead, they focus on maintaining height balance during insertions and deletions, ensuring the tree remains balanced enough to guarantee logarithmic search times.

For example, an AVL tree strictly maintains balance factors of -1, 0, or 1 at every node, which means its height is always kept near the minimum possible. This guarantees search, insertion, and deletion operations occur in O(log n) time consistently, whereas OBSTs provide optimal search times only when access probabilities are known and static.

Red-Black trees offer a more relaxed balancing rule, which usually results in faster insertions and deletions than AVL trees with slightly less strict height guarantees. Both are used extensively in database indexing engines and C++ STL map implementations (like std::map).

They’re a good fit if the access patterns change frequently or if you cannot predict key access probabilities accurately. Say a stockbroker’s order book system where new orders come and go with no fixed probabilities; balanced trees provide stable performance without costly recalculations.

Hashing and Its Tradeoffs

Hashing is another popular alternative to OBSTs, especially when the goal is quick lookups without the overhead of tree maintenance. Through hashing, keys are converted using a hash function to an index in an array, offering average-case constant time (O(1)) search performance.

Common hashing applications in finance include quick lookup tables for market data symbols or client account information. For instance, a trading platform might use a hash table to instantly retrieve user sessions or stock tickers without traversing a tree.

However, hashing does have its downsides. It doesn't maintain order among keys, which makes operations like range queries or in-order traversals inefficient or impossible. Also, hash collisions require collision resolution methods — chaining or open addressing — which can degrade performance if not handled well.

Memory overhead and hashing function design can be tricky too, as poorly chosen hash functions lead to clustering and slower accesses. OBSTs and balanced BSTs still hold advantage when you need sorted data or predictable performance without collision hazards.

Choosing the right data structure depends largely on your specific application's needs, including how predictable your access patterns are, whether you require ordered data processing, and how often your dataset changes.

In summary, while OBSTs shine when you know exactly how often each key will be accessed and don't need frequent updates, balanced BSTs are your go-to for more dynamic datasets, and hashing is unbeatable for raw speed in direct-access lookups without ordering concerns.

Implementing OBST in Programming

Understanding how to implement an optimal binary search tree (OBST) in programming is a vital step for those looking to make their data structures more efficient, especially when frequent searches with known probabilities are involved. Writing the OBST algorithm correctly not only enables a reduction in expected search time but also sharpens one's grasp of dynamic programming and recursive optimization techniques.

When it comes to practical usage, programmers often need to translate mathematical formulas and recurrence relations into code. This transformation requires a careful approach to handling probabilities, indexing, and memory management. It's not as straightforward as coding a regular binary search tree where the insertion order determines shape; OBST demands calculation up front to decide on the structure that minimizes search costs.

Algorithm Outline in Popular Languages

The core of an OBST implementation involves three main components: calculating the cost matrix, determining the root matrix, and constructing the tree itself. Let’s take a quick look at how these can be implemented in two frequently used programming languages — Python and Java.

• Python: Python's list comprehensions and dynamic typing make it relatively easy to handle the matrices. For instance, using nested lists for cost and root tables coupled with straightforward loops keeps the code readable. The recursive tree-building function then neatly uses these tables to assemble the tree nodes.

• Java: In Java, defining classes for tree nodes is customary, and arrays must be carefully managed for the cost and root tables. Java’s static typing means additional care must be taken to handle edge cases and indexing properly, but it offers robustness in larger applications.

Here is a very simplified snippet in Python, just to illustrate the calculation part:

python keys = [10, 12, 20] probabilities = [0.2, 0.5, 0.3]

Initialize tables

cost = [[0]*len(keys) for _ in keys] root = [[0]*len(keys) for _ in keys]

Base cases

for i in range(len(keys)): cost[i][i] = probabilities[i] root[i][i] = i

Filling tables using dynamic programming (simplified)

for length in range(2, len(keys)+1): for i in range(len(keys)-length+1): j = i + length - 1

Initialize min cost