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Each target is represented by a one-dimensional segment defined by its starting position and ending position along the track. A dart can be thrown at any specific position along the track, and it travels perfectly vertically, popping any target whose segment includes that position.\n\nMore formally, you are given a 2D integer array **intervals** where **intervals[i] = [start\u003csub>i\u003c/sub>, end\u003csub>i\u003c/sub>]** represents a target spanning from position **start\u003csub>i\u003c/sub>** to position **end\u003csub>i\u003c/sub>** (inclusive). A dart thrown at position **x** pops all targets where **start\u003csub>i\u003c/sub> ≤ x ≤ end\u003csub>i\u003c/sub>**.\n\nYour goal is to determine the **minimum number of darts** required to pop every single target. There is no limit on how many darts you can use, and each dart pops all targets along its vertical path.\n\nThis is fundamentally an interval coverage optimization problem: find the minimum number of points (dart positions) such that every interval contains at least one of the chosen points.","url":"https://www.onenoughtone.com/visualizers/minimum-darts-to-pop-intervals","learningResourceType":"interactive visualization","educationalUse":"instruction","provider":{"@type":"Organization","name":"OneNoughtOne","url":"https://www.onenoughtone.com"},"isAccessibleForFree":true,"inLanguage":"en"}6b:T446,Imagine a carnival game where floating targets are arranged along a horizontal track. Each target is represented by a one-dimensional segment defined by its starting position and ending position along the track. A dart can be thrown at any specific position along the track, and it travels perfectly vertically, popping any target whose segment includes that position. More formally, you are given a 2D integer array **intervals** where **intervals[i] = [start_i, end_i]** represents a target spanning from position **start_i** to position **end_i** (inclusive). A dart thrown at position **x** pops all targets where **start_i ≤ x ≤ end_i**. Your goal is to determine the **minimum number of darts** required to pop every single target. There is no limit on how many darts you can use, and each dart pops all targets along its vertical path. This is fundamentally an interval coverage optimization problem: find the minimum number of points (dart positions) such that every interval contains at least one of the chosen points.16:["$","div",null,{"className":"h-screen w-screen overflow-hidden","children":[["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https://www.onenoughtone.com/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Visualizers\",\"item\":\"https://www.onenoughtone.com/visualizers\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Minimum Darts to Pop Intervals\",\"item\":\"https://www.onenoughtone.com/visualizers/minimum-darts-to-pop-intervals\"}]}"}}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"$4a"}}],["$","$L6a",null,{"state":{"mutations":[],"queries":[{"dehydratedAt":1792199656407,"state":{"data":{"json":{"archivedAt":null,"id":"cmv9ua5490885li01ox2kw1zt","problemId":"D0452","slug":"minimum-darts-to-pop-intervals","title":"Minimum Darts to Pop Intervals","difficulty":"MEDIUM","problemType":"DSA","points":20,"description":"$6b","examples":[{"input":"points = [[10,16],[2,8],[1,6],[7,12]]","output":"2","explanation":"All targets can be popped with just 2 darts:\n- Throw a dart at position x = 6, which pops targets [2,8] and [1,6] since 6 lies within both intervals.\n- Throw a dart at position x = 11, which pops targets [10,16] and [7,12] since 11 lies within both intervals.\nThus, the minimum number of darts required is 2."},{"input":"points = [[1,2],[3,4],[5,6],[7,8]]","output":"4","explanation":"No two targets share any overlapping positions. Each target forms a completely disjoint interval:\n- [1,2] and [3,4] are separated (gap between 2 and 3)\n- [3,4] and [5,6] are separated (gap between 4 and 5)\n- [5,6] and [7,8] are separated (gap between 6 and 7)\nSince no single dart position can pop multiple targets, we need exactly 4 darts—one for each target."},{"input":"points = [[1,2],[2,3],[3,4],[4,5]]","output":"2","explanation":"Adjacent targets share boundary positions, allowing efficient coverage:\n- A dart at x = 2 pops targets [1,2] and [2,3] (2 is the right boundary of the first and left boundary of the second).\n- A dart at x = 4 pops targets [3,4] and [4,5] (4 is the shared boundary).\nWith strategic placement at shared boundaries, only 2 darts are needed to pop all 4 targets."}],"constraints":["1 ≤ intervals.length ≤ 10⁵","intervals[i].length == 2","-2³¹ ≤ start_i < end_i ≤ 2³¹ - 1"],"hints":["Think about how overlapping intervals can be covered by a single point. If you process intervals in a specific order, you can make greedy decisions about where to place each dart.","Sorting is key to this problem. Consider sorting intervals by their ending position. Why might the ending position be more useful than the starting position for making greedy choices?","After sorting by end position, use a greedy approach: place a dart at the rightmost valid position of the current interval's coverage. This position is the end point of the interval. Then skip all subsequent intervals that this dart covers.","The greedy insight: by placing a dart at the end of the current interval, you maximize the chance of it also covering subsequent intervals (when sorted by end). If the next interval starts after your dart position, you need a new dart."],"topics":["Array","Greedy","Sorting","Intervals"],"relatedProblems":["D0435","D0056","D0252"],"likes":0,"dislikes":0,"acceptedSubmissions":0,"totalSubmissions":0,"editorialContent":null,"followUpQuestions":[{"hint":"This becomes more complex as the greedy choice changes. You might need to prioritize which targets to pop with each dart based on their overlap patterns. Consider a modified greedy where at each dart position, you select the K targets that provide maximum benefit (e.g., those with earliest end times among overlapping ones).","question":"What if each target has a 'toughness' value and your darts have limited 'power'—each dart can pop at most K targets at its position. How would you modify the algorithm to minimize darts under this constraint?"},{"hint":"This is a different optimization problem. Consider dynamic programming or a weighted interval scheduling approach. Sort by end time and use DP where dp[i][j] represents the maximum targets popped using j darts considering the first i intervals.","question":"Instead of minimizing darts, suppose you have exactly K darts available. What's the maximum number of targets you can pop? How does this change the problem structure?"},{"hint":"Consider interval tree or segment tree data structures that support efficient insertions and deletions. Maintain sorted order of endpoints and track overlap counts. The answer relates to the maximum depth of interval overlaps, which can be maintained dynamically.","question":"How would you solve this problem if the intervals were dynamic—targets are continuously being added and removed—and you need to maintain the minimum dart count in real-time?"},{"hint":"The 2D version is significantly harder (NP-hard in general). For circular regions, you might use techniques like Helly's theorem for intersections, or heuristics like clustering algorithms. Exact solutions might require integer linear programming or branch-and-bound approaches.","question":"If the problem extends to 2D (circular targets on a wall instead of segments on a line), where each dart is a point that must fall within a circle to pop it, how would the approach change?"},{"hint":"This becomes a weighted set cover variant. You'd need to consider all potential dart positions (interval intersection points and endpoints) as candidates. Dynamic programming or integer programming approaches may be necessary depending on the cost structure.","question":"What if there's a cost associated with throwing a dart at each position (e.g., some positions are harder to reach), and you want to minimize total cost instead of dart count?"},{"hint":"Consider partitioning intervals by position ranges. Each partition can be solved independently, but boundary intervals might overlap across partitions. A merge step would reconcile boundary decisions. MapReduce-style processing with careful handling of partition boundaries could work.","question":"How would you parallelize this algorithm for processing billions of intervals across a distributed system? What are the synchronization challenges?"},{"hint":"During the greedy traversal, at each step there may be a range of valid positions for the dart (the overlap region of covered intervals). Track these ranges. To enumerate all solutions, use backtracking where at each choice point, you explore all valid positions within the overlap range.","question":"If you need to actually output the optimal dart positions (not just the count), and there are multiple optimal solutions, how would you enumerate all possible optimal dart configurations?"},{"hint":"First solve the problem for high-priority targets alone. Then greedily extend dart coverage to include as many low-priority targets as possible without adding new darts. If additional coverage is desired, add darts for remaining uncovered low-priority targets.","question":"What if intervals can have different priorities, and your goal is to minimize darts while ensuring all high-priority targets are definitely popped, but low-priority targets are optional?"}],"realWorldApplications":[{"title":"Network Broadcast Optimization","industry":"Telecommunications & Networking","description":"In wireless networks and broadcast systems, base stations need to transmit messages to devices within their range. Each device has a coverage window (range of tower positions that can reach it). Network engineers use interval covering algorithms to determine the minimum number of broadcast points (tower positions or transmission times) needed to reach all devices. This directly reduces infrastructure costs and power consumption in cellular networks, WiFi deployments, and IoT sensor networks. Companies like Qualcomm and Ericsson employ these algorithms in their network planning tools."},{"title":"Automated Testing and Quality Assurance","industry":"Software Engineering & DevOps","description":"In software testing, especially for embedded systems and APIs, test cases often cover ranges of input values. The goal is to select the minimum number of test inputs that collectively cover all specified ranges of behavior. This interval covering approach is used in test suite minimization, where QA teams reduce thousands of tests to a minimal covering set without sacrificing coverage. This saves significant regression testing time in CI/CD pipelines at companies like Google, Microsoft, and major automotive software suppliers."},{"title":"Vaccine and Medication Distribution Centers","industry":"Healthcare & Public Health","description":"When planning vaccination or medication distribution, health authorities consider geographic service ranges—each distribution center can serve people within a certain travel distance or time window (represented as an interval on a map projection). Public health planners use interval covering algorithms to determine the minimum number of distribution sites needed to ensure all population segments have access. This was crucial during COVID-19 vaccine rollouts where officials optimized limited distribution resources across populations with varying accessibility constraints."},{"title":"Video Surveillance Camera Placement","industry":"Security & Surveillance","description":"Security systems for warehouses, retail stores, and smart cities need to monitor all areas with the fewest cameras possible. Each potential camera position covers a range of the monitored space (represented as intervals in a 1D projection or approximation). Security architects use covering algorithms to find the minimum camera count that eliminates all blind spots. 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Tools like Eventbrite, Cvent, and university scheduling systems incorporate these algorithms to minimize venue hours while accommodating all participants."},{"title":"Satellite Downlink Scheduling","industry":"Aerospace & Space Operations","description":"Earth observation satellites (used by NASA, ESA, and commercial providers like Planet Labs) orbit the Earth and can only downlink data to ground stations during specific time windows when they pass overhead. Each ground station has availability intervals, and mission planners must schedule the minimum number of downlink sessions (across multiple ground stations) to retrieve all collected data. 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