Algorithm 1: HNSW graph generation and search overview.
▷ Global Variables:
1: trainingDataset;	{Complete KNN training dataset}
2: hnswGraph;	{HNSW multilayer graph}
3: topLayer;	{Maximum, top layer of the hnswGraph}
4: entryPoint;	{Entry point for the hnswGraph}
5: mL;	{Normalization factor for level definition}
6: ef;	{Number of nearest neighbors to use}
▷ Function:
7: hnswGraphGeneration()	{Generate hnswGraph from trainingDataset}
Action:
8: for each element newElement from trainingDataset do
9:    maxLayer ← ⌊−ln(uniform(0 ... 1)) × mL⌋;	{Compute the maximum layer of the element}
10:    if entryPoint = = nil then	{First inserted element.}
11:      topLayer ← maxLayer;
12:      entryPoint ← newElement;	{Sets the entry point for the hnswGraph}
13:      next;
14:    end if
{Search the nearestNode already in hnswGraph starting from the topLayer until maxLayer}
15:    nearestNodes = entryPoint;
16:    for currentLayer ← topLayer ... maxLayer do
17:      nearestNodes ← searchLayer(newElement, nearestNodes, currentLayer);
18:    end for
{Insert the newElement in all of the layers from maxLayer to layer 0}
19:    for currentLayer ← maxLayer ... 0 do
20:      nearestNodes ← searchLayer(newElement, nearestNodes, currentLayer);
21:      neighbors ← selecNeighbors(newElement, nearestNodes, currentLayer);
22:      insertElement(newElement, currentLayer, neighbors);
23:    end for
24:    if maxLayer > topLayer then
25:      topLayer ← maxLayer;
26:      entryPoint ← newElement;	{Set entry point for hnswGraph to newElement}
27:    end if
28: end for
▷ Function:
29: hnswSearch(element)	{Search nearest nodes for element from trainingDataset}
Action:
30: {Search the element in hnswGraph starting from the topLayer until 0}
31: nearestNodes = entryPoint;
32: for currentLayer ← topLayer . . . 0 do
33:    nearestNodes ← searchLayer(newElement, currentLayer, nearestNodes);
34: end for
35: return nearestNodes;
▷ Function:
36: searchLayer(element, nearestNodes, layer)	{Search the ef nearest nodes for element in layer}
37:	{using nearestNodes as enter points.}
Action:
38: nearestNodes ← {Greedy graph search algorithm as described in [16]}
39: return nearestNodes;