AI Engineering @ECEnggNotes

Heaps power priority queues and heap sort. They provide efficient access to extremal elements without maintaining full sorted order. The partial ordering property is sufficient for many scheduling and selection algorithms.

Both insertion and extraction run in O(log n) time. The height of a complete binary tree is logarithmic. Each bubble operation traverses at most one level per comparison. Finding the minimum is O(1) since it always occupies index zero.

Array-based implementation eliminates pointer overhead. The complete tree structure maps cleanly to contiguous memory. Dynamic resizing follows the ArrayList pattern: double capacity when full and copy elements over.

A heap is a complete binary tree that maintains a specific ordering property. In a min-heap, each node is smaller than its children. The smallest element always sits at the root. Max-heaps invert this relationship.

Heaps are implemented using arrays, not node objects. Elements fill sequentially with no gaps. This allows direct index calculation instead of pointer references. For an element at index i: left child is at 2i+1, right child at 2i+2, parent at (i-1)/2.

Removing the minimum element requires extracting the root. Replace it with the last element in the heap. Then bubble down: compare the new root with both children and swap with the smaller child. Repeat until the element finds its correct position.

Insertion follows a two-step process. The new element goes into the next available position, filling the tree level by level from left to right. Then it bubbles up by comparing with its parent. If out of order, swap and repeat until the heap property is restored.

As temperature rises, the energy levels available for the molecule to occupy also increase. Hence the *probability* of predicting the *energy states* occupied by the molecule decreases. An increased number of available energy states is indicative of the *disorder* in the system.

Energy states of atomic particles are like shelves where you stack items. No item exists in between the shelves.

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Any spontaneous change – that is, any change capable of producing work – is also accompanied by an increase in the entropy in the universe. We call this an entropy tax.

Consider a closed system like a cylinder with a piston. Heat Tax : A chemical reaction produces gases whose volume exceeds that of the reactants. The system uses the heat from the chemical reaction to push the piston outwards. Heat is converted to work.

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Helmholtz free energy and Gibbs free energy are crucial for determining the spontaneity of a process and represent the energy available to do work under specific conditions.

In thermodynamics, Helmholtz free energy and Gibbs free energy are thermodynamic potentials that measure the "useful" work obtainable from a closed thermodynamic system. What is a closed thermodynamic system?