The “Furthest Building You Can Reach” problem asks you to determine how far you can progress along a sequence of building heights using a limited number of bricks and ladders. To move from one building to the next, you must climb the height difference if the next building is taller. You can pay that cost using either bricks, which are consumed equal to the height difference, or ladders, which can cover any height difference for free. The goal is to reach the furthest possible building index.
This problem is not about summing heights or simulating all possibilities. It is about resource allocation. You have two types of resources with very different properties, and how you assign them across climbs determines how far you can go.
Why greedy but naive strategies fail
A tempting idea is to always use ladders for the earliest climbs or to always use bricks first and save ladders for later. Both strategies can fail. Using a ladder too early on a small climb wastes its power, while using bricks on a very large climb can drain your resources prematurely.
The difficulty is that you do not know which climbs are “worth” a ladder until you see the relative sizes of all climbs encountered so far. This makes the problem unsuitable for simple one-pass greedy rules without some form of adjustment.
Understanding the key greedy insight
The core insight is that ladders are most valuable when used on the largest climbs. Bricks, on the other hand, are best spent on smaller climbs because their cost scales directly with height difference.
This suggests a greedy strategy with a twist: as you encounter climbs, you tentatively assign ladders to them, but you remain willing to change your mind if you later encounter a larger climb. In other words, you want ladders to end up covering the biggest height differences among all climbs you actually take.
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Using a priority structure to manage climbs
To implement this idea, you track the height differences of all upward moves you have encountered. Conceptually, every time you climb up, you assume you will use a ladder for that climb and record its height difference.
If the number of climbs you have recorded exceeds the number of ladders available, you must replace one ladder usage with bricks. To minimize brick usage, you should replace the ladder that was covering the smallest climb so far, because bricks are cheapest there. This swap frees the ladder to potentially cover a larger climb later.
This “assign ladders to the largest climbs” principle is the heart of the solution.
Why this greedy adjustment works
The correctness of this approach comes from an exchange argument. Suppose you used a ladder on a small climb and bricks on a larger one. Swapping those assignments would reduce or maintain brick usage and never make things worse. Therefore, in an optimal allocation, ladders must be assigned to the largest climbs encountered so far.
By continually enforcing this rule as you progress, you ensure that at any point, your resource usage is as efficient as possible, given the buildings you have reached.
Detecting when you can no longer proceed
As you process buildings one by one, you deduct bricks whenever a ladder assignment is replaced. If at any point your brick count drops below zero, it means you do not have enough resources to cover the required climbs up to this building. At that moment, you cannot proceed further, and the current building index is the furthest you can reach.
This stopping condition is simple and reliable because resource usage is always kept minimal under the greedy strategy.
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Why this approach guarantees the furthest reach
By always minimizing brick consumption for the climbs you have already encountered, you maximize the chance of surviving future climbs. Any other strategy that uses bricks less efficiently would run out sooner.
The algorithm does not try to predict the future. Instead, it dynamically adjusts past decisions to ensure ladders are reserved for the most expensive climbs seen so far. This adaptability is what makes the greedy strategy correct.
Time and space complexity considerations
The algorithm processes each building once and maintains a structure of encountered climbs. Operations on this structure are efficient and scale with the number of ladders rather than the number of buildings.
Space usage is limited to storing climb differences. Overall, the solution runs efficiently even for large inputs.
Why this problem is common in interviews
The Furthest Building You Can Reach is popular in interviews because it tests whether candidates understand greedy algorithms beyond simple rules. The need to revise earlier decisions is subtle and often overlooked.
It also tests whether candidates can reason about resource allocation and use the right abstraction to support greedy adjustments.
What this problem teaches beyond buildings
Beyond its narrative, this problem teaches a powerful algorithmic lesson: when you have limited high-power resources and unlimited low-power resources, the high-power ones should be reserved for the most expensive tasks. Implementing this principle often requires tracking past decisions and being willing to revise them.
If you can clearly explain why ladders belong on the largest climbs, how brick usage is minimized dynamically, and why running out of bricks marks the stopping point, you demonstrate strong greedy reasoning. That depth of understanding makes “Furthest Building You Can Reach” an excellent exercise in adaptive greedy algorithms and resource optimization.
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