https://github.com/facebookresearch/dino/tree/main

https://juejin.cn/post/7224738994825789496
https://www.youtube.com/watch?v=h3ij3F3cPIk&t=1005s
DI+NO（蒸馏+No Label）
具体来说，DINO 是使用一种称为“无监督自蒸馏”的方法，该方法通过自监督学习来学习模型的知识表示。在这个方法中，模型使用自身的输出来生成“伪标签”，然后使用这些伪标签来重新训练模型，从而进一步提高模型的性能和泛化能力。

知识蒸馏

https://blog.csdn.net/xbinworld/article/details/83063726

重点idea就是提出用soft target来辅助hard target一起训练，而soft target来自于大模型的预测输出。这里有人会问，明明true label（hard target）是完全正确的，为什么还要soft target呢？
hard target 包含的信息量（信息熵）很低，soft target包含的信息量大，拥有不同类之间关系的信息（比如同时分类驴和马的时候，尽管某张图片是马，但是soft target就不会像hard target 那样只有马的index处的值为1，其余为0，而是在驴的部分也会有概率。）[5]
这样的好处是，这个图像可能更像驴，而不会去像汽车或者狗之类的，而这样的soft信息存在于概率中，以及label之间的高低相似性都存在于soft target中。但是如果soft targe是像这样的信息[0.98 0.01 0.01]，就意义不大了，所以需要在softmax中增加温度参数T（这个设置在最终训练完之后的推理中是不需要的）

ViT

DINO

关于DINO中发生的涌现
https://juejin.cn/post/7280436457142501388

DINO之前的工作

We have also seen emerged two properties that can be leveraged in future applications: the quality of the features in k-NN classification has a potential for image retrieval. The presence of information about the scene layout in the features can also benefit weakly supervised image segmentation.

https://blog.csdn.net/h661975/article/details/135116957

loss: ITC (Image Text Contrastive)

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# image_encoder - ResNet or Vision Transformer 
# text_encoder - CBOW or Text Transformer 
# I[n, h, w, c] - minibatch of aligned images 
# T[n, l] - minibatch of aligned texts 
# W_i[d_i, d_e] - learned proj of image to embed 
# W_t[d_t, d_e] - learned proj of text to embed 
# t - learned temperature parameter  

# extract feature representations of each modality 
I_f = image_encoder(I) #[n, d_i] 
T_f = text_encoder(T) #[n, d_t]  

# joint multimodal embedding [n, d_e] 
I_e = l2_normalize(np.dot(I_f, W_i), axis=1) T
_e = l2_normalize(np.dot(T_f, W_t), axis=1)  

# scaled pairwise cosine similarities [n, n] 
logits = np.dot(I_e, T_e.T) * np.exp(t)  

# symmetric loss function 
labels = np.arange(n) 
loss_i = cross_entropy_loss(logits, labels, axis=0) 
loss_t = cross_entropy_loss(logits, labels, axis=1) 
loss = (loss_i + loss_t)/2

Cross_entropy_loss:

CLIP 本质上是全局图像嵌入，不利于像素对齐特征提取。

NeRF+CLIP

Intro

背景

• 神经辐射场 (NeRF) 已成为一种强大的技术，用于捕获复杂的现实世界 3D 场景的逼真数字表示。然而，NeRF 的直接输出只不过是一个彩色的密度场，缺乏意义或上下文，这阻碍了构建与生成的 3D 场景交互的界面。
• 自然语言是与 3D 场景交互的直观界面。考虑厨房的捕获。想象一下，能够通过询问“用具”在哪里来导航这个厨房，或者更具体地说，询问可用于“搅拌”的工具，甚至可以询问您最喜欢的带有特定功能的杯子。其上的徽标——贯穿日常对话的舒适和熟悉。这不仅需要处理自然语言输入查询的能力，还需要能够在多个尺度上合并语义并与长尾和抽象概念相关。

解决方案

一个Language Field
通过优化从现成的视觉语言模型（如 CLIP）到 3D 场景的嵌入，为 NeRF 中的语言奠定基础。
LERF 提供了一个额外的好处：由于我们从多个尺度的多个视图中提取 CLIP 嵌入，因此通过 3D CLIP 嵌入获得的文本查询的相关性图与通过 2D CLIP 嵌入获得的文本查询的相关性图相比更加本地化。根据定义，它们也是 3D 一致的，可以直接在 3D 字段中进行查询，而无需渲染到多个视图。

相较于Clip-Field[[CLIP-Fields- Weakly Supervised Semantic Fields for Robotic Memory]], LERF 更密集。

CLIP-Fields [32] and NLMaps-SayCan [8] fuse CLIP embeddings of crops into pointclouds, using a contrastively supervised field and classical pointcloud fusion respectively. In CLIP-Fields, the crop locations are guided by Detic [40]. On the other hand, NLMaps-SayCan relies on region proposal networks. These maps are sparser than LERF as they primarily query CLIP on detected objects rather than densely throughout views of the scene. Concurrent work ConceptFusion [19] fuses CLIP features more densely in RGBD pointclouds, using Mask2Former [9] to predict regions of interest, meaning it can lose objects which are out of distribution to Mask2Former’s training set. In contrast, LERF does not use region or mask proposals.

LERF

给定一组校准的输入图像，我们将 CLIP 嵌入到 NeRF 内的 3D 场中。然而，查询单个 3D 点的 CLIP 嵌入是不明确的，因为 CLIP 本质上是全局图像嵌入，不利于像素对齐特征提取。为了解释这一特性，我们提出了一种新颖的方法，该方法涉及学习以样本点为中心的卷上的语言嵌入领域。具体来说，该字段的输出是包含指定体积的图像作物的所有训练视图中的平均 CLIP 嵌入。通过将查询从点重新构造为体积，我们可以有效地从输入图像的粗裁剪中监督密集的字段，这些图像可以通过在给定的体积尺度上进行调节来以像素对齐的方式渲染。

https://blog.csdn.net/amusi1994/article/details/129701012

主要记录一些读场景语义化重建的论文的过程中的想法

重要的问题

多模态包含哪些任务

• 图文检索 Image-text Retrival
• 视觉问答 VQA
• 视觉推理 Visual Reasoning
• 视觉蕴含 Visual Entailment

多模态有哪些loss

• Image Text Contrastive(ITC) [[CLIP多模态预训练模型]]
• Word Patch Aligment (WPA) used in object detection ViT
• Image Text Matching (ITM)
• Mask Languae Modeling (MLM) BERT 完形填空

给定一个具体的任务，机器人需要哪些场景信息才能顺利执行这个任务（通用机器人）

限定：暂不考虑机器人的移动性，也就是不需要跨视野的导航(OK-Robot)，暂定为桌面机器人

具体来说，通用机器人的特点包括：

1. 多任务能力：能够执行多种不同类型的任务，如装配、搬运、清洁、检测等。
2. 适应性强：具备适应多种环境和工作条件的能力，例如在不同地形或生产线中工作的能力。
3. 智能控制：通过先进的传感器、人工智能算法、机器学习技术等手段，能够实现自主决策和任务规划。

• 物体的具体形状（用于抓取, grab-anything）
• 物体语义信息（grounded caption, clip）

Recognize The Relationships Between Child & Parent

受DINO自蒸馏自监督的启发，可以通过物体活动的图像序列来推测物体各个部分的物理关系(attention map)[[DINO]]

训练集可以使用Unity生成不同的光影/物体，连接语义

Build the physics world in robot mind

voxel collider for detected objects, joints, physics agent interact with physics engine.
点云数据，grounded caption=>object property, hierarchy relation, joints(maybe new model should be proposed)

语义还原物体模型

受[[BLIP]]启发，understanding for language & existing point cloud, generation for the rest of the point cloud (Wonder3D已实现)

疑问：

• 和LERF [[LERF- Language Embedded Radiance Fields]] 的区别

是什么

A spatial-semantic memory
是一个隐式场景模型，可用于各种任务，例如分割、实例识别、空间语义搜索和视图定位
CLIP-Fields 学习从空间位置到语义嵌入向量的映射。
这种映射可以仅通过来自网络图像和网络文本训练模型（例如 CLIP[[CLIP多模态预训练模型]]、Detic 和 Sentence-BERT）的监督进行训练；因此不使用直接的人类监督。

基于的工作

• CLIP [[CLIP多模态预训练模型]] : 基于训练一对图像和语言嵌入网络，使得图像和描述该图像的文本字符串具有相似的嵌入。 在这项工作中大量使用 CLIP 模型和嵌入，因为它们可以作为对象的视觉特征及其可能的语言标签之间的共享表示。
• Detic : 开放标签对象检测和图像分割，允许用户在运行时定义标签集，无需额外的训练或微调。用于生成数据集
• Sentence-BERT : 用于文本相似性的句子嵌入网络
• Instant-NGP : 构建了从空间（可能还有时间）坐标到某些物理属性的映射，例如神经辐射场情况下的 RGB 颜色和密度，或即时有符号距离场情况下的有符号距离

方法

Goal

We aim to build a system that can connect points of a 3D scene with their visual and semantic meaning.
Provide an interface with a pair of scene-dependent implicit functions $f, h : R^3 → R^n$ such that for the coordinates of any point P in our scene, f (P ) is a vector representing its semantic features, and h(P ) is another vector representing its visual features.

Dataset creation

貌似每针对一个新场景都需要重新train一遍来获得坐标到语义的映射。

Intro

Creating a general-purpose robot has been a longstanding dream of the robotics community.

背景

当前想要实现这一目标的系统脆弱、封闭，并且在遇到未见过的情况时会失败。即使是最大的机器人模型通常也只能部署在以前见过的环境中 [5, 6]。在机器人数据很少的环境中，例如在非结构化的家庭环境中，这些系统的脆弱性会进一步加剧。

虽然大型视觉模型显示出语义理解 、检测以及将视觉表示与语言联系起来的能力并且与此同时，机器人的导航、抓取和重新排列等基本机器人技能已经相当成熟。
但是将现代视觉模型与机器人特定基元相结合的机器人系统表现非常差。

这可能是因为单纯将多个不确定性的系统组合在一起会导致准确率急剧恶化。
所以我们需要一个将VLM和机器人primitives(导航，抓取，放置)结合在一起的细致框架，即OK-Robot。

发现

• 预训练的 VLM 对于开放词汇导航非常有效: 当前的开放词汇视觉语言模型，例如 CLIP 或 OWL-ViT，在识别现实世界中的任意对象方面提供了强大的性能，并能够以零样本的方式导航到它们。
• 预训练的抓取模型可以直接应用于移动操作：与 VLM 类似，经过大量数据预训练的专用机器人模型可以立即应用于家庭中的开放词汇抓取。这些机器人模型不需要任何额外的训练或微调。
• 如何组合组件至关重要：给定预训练模型，我们发现可以使用简单的状态机模型将它们组合在一起，无需训练。我们还发现，使用启发式方法来抵消机器人的物理限制可以在现实世界中获得更高的成功率。
• 仍然存在一些挑战：虽然，考虑到在任意家庭中进行零样本的巨大挑战，OK-Robot 在之前的工作基础上进行了改进，通过分析故障模式，我们发现 VLM、机器人模型和机器人形态可以进行重大改进，这将直接提高开放知识操纵代理的性能。

Methodology

该框架主要完成的任务

Pick up A (from B) and drop it on/in C”, where A is an object and B and C are places in a real-world environment such as homes

Open-home, open-vocabulary object navigation

负责空间重建，识别物体大致位置，机器人导航
用到的方法:

• CLIP-Fields [[CLIP-Fields- Weakly Supervised Semantic Fields for Robotic Memory]] : a RGB-D video of the home -> a sequence of posed ( with camera pose and positions) RGB-D images，用于重建环境，该研究还基于此获取了环境中物体和容器旁边的地板表面。
• OWL-ViT [[Simple Open-Vocabulary Object Detection with Vision Transformers]] : 我们在每一帧上应用检测器，并提取每个对象边界框、CLIP-embedding、检测器置信度，并将这些信息传递到object memory模块中
• SAM: 用于将ViT的检测框转化为mask
• VoxcelMap: similar to object-centric memory of CLIP-Fields [[CLIP-Fields- Weakly Supervised Semantic Fields for Robotic Memory]], 基于点云中每一个点的CLIP semantic vector,每一个5cm的体素都包含一个CLIP-embedding的detector-confidence weighted average.
• Querying the memory module: 先将language query 转化成CLIP semantic vector,然后基于voxelmap的clip-embeding，寻找最语义接近的那个voxel，以此定位。
• 

Experiment

Introduction

IL是区别于传统手动编程来赋予机器人自主能力的方法。
IL 允许机器通过演示（人类演示专家行为）来学习所需的行为，从而消除了对显式编程或特定于任务的奖励函数的需要。
IL主要有两个类别：

• 行为克隆(BC)
• 反向强化学习(IRL)
  

Behavior Cloning

BC 是一种 IL 技术，它将学习行为的问题视为监督学习任务 。 BC 涉及通过建立环境状态与相应专家操作之间的映射来训练模型来复制专家的行为。专家的行为被记录为一组state-action pair，也称为演示。在训练过程中，模型学习一个函数，利用这些演示作为输入，将当前状态转换为相应的专家操作。经过训练，模型可以利用这个学习函数来生成遇到新状态的动作。

不需要了解环境的潜在动态，计算效率很高，相对简单的方法。

The covariate shift problem: 测试期间观察到的状态分布可能与训练期间观察到的状态分布有所不同，使得代理在遇到未见过的状态时容易出错，而对于如何进行操作缺乏明确的指导。BC监督方法的问题是，当智能体漂移并遇到分布外状态时，它不知道如何返回到演示的状态。

为了解决这个问题：

Inverse Reinforcement Learning

IRL 涉及一个学徒代理，其任务是推断观察到的演示背后的奖励函数，这些演示被认为源自表现最佳的专家 。然后使用推断的奖励函数通过 RL 训练学习代理的策略。

为了解决“政策->奖励函数“的模糊性，有以下三种IRL

• maximum-margin methods（奖励函数比任何其他策略在一定程度上更全面地解释最优策略。这本质上意味着找到一个最大化指定利润的解决方案，确保派生的奖励函数捕捉专家行为的本质。）
• maximum entropy（处理专家次优性和随机性的有前景的能力）
• guided cost learning（旨在优化策略优化内循环内的非线性奖励函数的方法。这种方法通过直接利用系统的原始状态来构建奖励函数，从而改变了传统的 IRL 范式，从而消除了广泛的特征工程的需要。）

Adversarial Imitation Learning

The agent strives to deceive the discriminator by generating trajectories closely resembling those of the expert.

Imitation From Observation

仅通过图像序列来学习，不需要具体的关节动作操作数据。

Unlike the traditional methods, IfO presents a more organic approach to learning from experts, mirroring how humans and animals approach imitation. Humans often learn new behaviors by observing others without detailed knowledge of their actions (e.g., the muscle commands). People learn a diverse range of tasks, from weaving to swimming to playing games, by watching online videos. Despite differences in body shapes, sensory inputs, and timing, humans exhibit an impressive ability to apply knowledge gained from the online demonstrations

将可学习的资源扩大到了线上的视频资源。

Latent Action Policies (LAPOs)

过分析观察到的动态，LAPO 推断出行动空间的底层结构，促进潜在行动策略的训练。然后，这些策略可以进行高效的微调，以达到专家级的性能，从而提供离线和在线场景的适应性。使用包含标记动作的小数据集进行离线微调是可行的，而在线微调可以使用奖励来完成。与依赖标记数据来训练逆动力学模型不同，LAPO直接从观察到的环境动态中导出潜在动作信息，而不需要任何标签。

Challenges And Limitations

。。。

Background

https://validator.w3.org/ : 是一个由万维网联盟（W3C）提供的在线工具，用于检查网页的 HTML、XHTML 或其他标记语言是否符合相关标准和规范。它可以帮助开发者提高网页的质量和兼容性，确保网页在不同浏览器和设备上正确显示。

作为实习生你需要完成网站开发，要求如下(不重要)

During the next few hours you are explained in details all the requirements you have to match. Among
the most emphasized points you learn that you must (i) use the last version of Microsoft Front Page
Express to write the websites; (ii) include as many buttons as possible (even if one is enough); (iii) when a
hidden box is expanded, do so as high as possible above the button which opened it and do not notify the
user; (iv) as much as possible do not disable or hide irrelevant information, simply include it the middle
of the useful content; (v) use and abuse pop ups; (vi) feel free to include Chinese text in the middle of
an English text; (vii) if a page includes videos, ensure they are all fully downloaded before the user can
do anything; You also learn that they pay much attention to the quality of their product, and as such
you should never forget to test your website, IE 6 being recommended.
在接下来的几个小时里，我们会详细解释您需要满足的所有要求。其中最受关注的要点是：您必须 (i) 使用最新版本的 Microsoft Front Page Express 来编写网站；(ii) 包含尽可能多的按钮（即使一个按钮就足够了）；(iii) 展开隐藏框时，尽可能将其展开到打开它的按钮上方，并且不要通知用户；(iv) 尽可能不要禁用或隐藏不相关的信息，只需将其包含在有用内容的中间；(v) 使用和滥用弹出窗口；(vi) 可以在英文文本中间随意添加中文文本；(vii) 如果页面包含视频，请确保在用户执行任何操作之前，它们都已完全下载；您还了解到他们非常重视产品质量，因此您永远不要忘记测试您的网站，建议使用 IE 6.

大概是马牛特有的讽刺。。。总之就是总结了一堆网站开发的禁止事项。
因为这些要求的存在（特别是网页开发环境只能在windows下使用），我们作为一位只有一台双系统电脑的实习生遇到了障碍：

每次开发网页我们都需要切换到windows系统并暂停linux系统下的任务。然而在linux下我们需要跑一些任务，需要尽可能早去完成。

针对这些linux任务，马牛貌似将其分成了T个独立的任务，根据output推测，每个任务也可以被暂停，然后切换到windows进行开发。

可以通过把任务转交给朋友的方式来从schedule中移除部分网页开发的任务，至多两个网页。每个网页有对应的用于交易的作业量，作业量加起来不能超过HC的触发阈值。

标识

需要注意的是，计算任务需要的时间 $t_i$ 和网页完成的deadline $t_i$ 不是一个东西，建议区分标记。

输入

计算任务，网页开发以及外援的信息

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P H
s_0 t_0 h_0 # info for each page
... # omit P-1 lines
T # number of tasks
t_0
... # omit T-1 lines

以下纯来自AI的solution

To solve this problem, you need to combine optimization, scheduling, and greedy strategies. Here’s a step-by-step guide:

1. Parse the Input

• Read:
  • P,H: Number of webpages and maximum allowable exercises for delegation.
  • Details for each webpage (s_i, t_i, h_i): Start time, deadline, and exercise cost.
  • T: Number of computational tasks and their durations (t_i).

2. Determine Delegation Strategy

Key Idea:

• Delegate up to 2 webpages to the friend such that the total exercise cost (h_i) is minimized and ≤H.

Steps:

1. Sort webpages by their exercise cost (h_i).
2. Select up to 2 webpages to delegate:
   • Choose pages with the smallest h_i that satisfy h_1+h_2≤H.
3. Remove these pages from your schedule, as they are handled by the friend.

3. Calculate Free Time Intervals

Key Idea:

• After delegating pages, compute the blocked time intervals caused by the remaining webpages.

Steps:

1. For all non-delegated webpages, construct intervals [s_i, t_i).
2. Merge overlapping intervals to avoid double-counting blocked time.
3. Identify the free intervals:
   • Initially, it is [0,∞).
   • Subtract merged blocked intervals to get the free time slots.

4. Schedule Computational Tasks

Key Idea:

• Fit each computational task into the free intervals to minimize its completion time.

Steps:

1. Sort computational tasks by their required time (t_i).
2. Iterate over each task:
   • Start placing the task in the earliest available free interval.
   • If the task fits, calculate its completion time and reduce the interval size.
   • If it doesn’t fit, move to the next free interval.
3. Store the completion time for each task.

5. Output Results

For each computational task, output the minimum completion time determined in the previous step.

Algorithm Complexity

1. Input Parsing: O(P+T)
2. Delegation:
   • Sorting webpages: O(P \log P)
   • Selecting 2 webpages: O(1)
3. Free Interval Calculation:
   • Merging intervals: O(P \log P)
4. Task Scheduling:
   • Sorting tasks: O(T \log T)
   • Scheduling: O(T×free intervals), typically O(T + P).

Overall complexity: O((P + T) \log (P + T)).
this complexity is acceptable regarding the given input range.

OCaml Implementation

Here’s a high-level pseudocode that can be adapted into OCaml:

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(* Input Parsing *)
let read_input () =
  let p, h = read_line () in
  let webpages = List.init p (fun _ -> read_line ()) in
  let t = read_int () in
  let tasks = List.init t (fun _ -> read_int ()) in
  (p, h, webpages, tasks)

(* Delegate webpages *)
let delegate_webpages webpages h =
  let sorted_pages = List.sort (fun (_, _, h1) (_, _, h2) -> compare h1 h2) webpages in
  let rec select_delegation remaining_pages acc h_left =
    match remaining_pages with
    | [] -> acc, remaining_pages
    | (s, t, h)::tl when h <= h_left -> select_delegation tl ((s, t, h)::acc) (h_left - h)
    | _ -> acc, remaining_pages
  in
  select_delegation sorted_pages [] h

(* Compute free intervals *)
let compute_free_intervals remaining_pages =
  let intervals = List.map (fun (s, t, _) -> (s, t)) remaining_pages in
  let merged_intervals = merge_intervals intervals in
  calculate_free_time_slots merged_intervals

(* Schedule tasks *)
let schedule_tasks free_intervals tasks =
  List.map (fun task ->
    find_earliest_completion free_intervals task
  ) tasks

(* Main function *)
let () =
  let p, h, webpages, tasks = read_input () in
  let delegated_pages, remaining_pages = delegate_webpages webpages h in
  let free_intervals = compute_free_intervals remaining_pages in
  let task_completions = schedule_tasks free_intervals tasks in
  List.iter print_int task_completions

以下纯来自AI的proof

Proof of Correctness for the Algorithm

The algorithm is designed to solve the problem of scheduling computational tasks on a single computer while meeting constraints related to webpage deadlines and delegation. Below is a formal proof of its correctness.

1. Problem Understanding

The problem involves:

1. Allocating time for webpage writing (either by handling them yourself or delegating to a friend).
2. Maximizing free intervals for computational tasks.
3. Minimizing the completion times of computational tasks.

To ensure correctness, we need to prove:

• Webpage constraints are respected.
• Free time intervals are accurately calculated.
• Computational tasks are scheduled optimally.

2. Proof Components

2.1 Webpage Delegation

• Key Idea: Delegate up to 2 webpages with the smallest exercise costs such that the total cost does not exceed H.
• Proof:
  • By sorting webpages by h_i (exercise cost) and selecting the top 2 that satisfy h1+h2≤H, the algorithm guarantees that:
    • The delegation respects the maximum exercise limit H.
    • The webpages with the least impact on constraints are delegated, maximizing the remaining free time.

2.2 Free Interval Calculation

• Key Idea: Compute free intervals by merging blocked intervals caused by non-delegated webpages.
• Proof:
  • Non-delegated webpages define intervals [s_i, t_i) that block the computer.
  • Merging overlapping intervals ensures no double-counting of blocked time, leaving only valid free intervals.
  • The algorithm iteratively merges intervals by sorting them by s_i and combining overlapping or adjacent intervals, which is a standard and proven approach.

2.3 Task Scheduling

• Key Idea: Fit each computational task into the earliest available free interval to minimize its completion time.
• Proof:
  • Tasks are processed in increasing order of required time (t_i), ensuring that smaller tasks are scheduled earlier, leaving larger free intervals for longer tasks.
  • For each task:
    • The algorithm scans free intervals and places the task in the earliest slot where it fits.
    • This greedy approach guarantees the earliest possible completion time for each task because:
      • Free intervals are disjoint and sorted.
      • A task placed in an earlier slot cannot complete earlier in any other configuration without violating disjointness.

2.4 Overall Algorithm

• Correctness Statement: The algorithm produces the earliest possible completion times for all tasks while satisfying webpage constraints and delegation limits.
• Proof:
  • Webpages: Constraints are respected by delegating up to 2 pages within H exercises and blocking the necessary intervals for non-delegated pages.
  • Free Intervals: Accurate calculation ensures no overlap or error in available time slots.
  • Task Scheduling: Greedy placement in the earliest available interval ensures optimal completion times.

4. Edge Cases

• P=0,T=0: No webpages or tasks, trivial solution.
• H=0: No delegation possible, all webpages must be handled by the user.
• Overlapping webpage intervals: Correctly merged to avoid double-counting.
• Tasks larger than any free interval: Scheduled in the earliest free interval where they fit.

5. Conclusion

The algorithm is correct as it:

1. Adheres to all constraints (webpage deadlines, delegation limits, task requirements).
2. Ensures optimal scheduling of computational tasks.
3. Handles edge cases gracefully.

Would you like a detailed explanation for a specific part of this proof or assistance implementing it?