## How could a key could be inserted in a heap without increasing the size of an array?

``MAX-HEAP-INSERT(A, key)     A.heap-size = A.heap-size + 1     A[A.heap-size] = -infinity     HEAP-INCREASE-KEY(A,A.heap-size,key) ``

How could a key could be inserted in a heap without increasing the size of an array? With this code from Introduction To Algorithms, you can’t just randomly increase the heap size upon wish. Did I miss something? All all online lectures I have seen do no talk about this issue. Neither does the book touch this issue. Or is it that the lowest key in an array would be dropped automatically?

## What’s the historical interaction between the Wish spell and increasing ability scores?

I vaguely recall Wishes in 1e affecting ability score increases going something like this:

• a single Wish spell could increase any ability score to 16
• you needed another Wish spell to get to 18
• then it was one Wish spell per point over that (with like 5 Wishes needed to get to 18/00 STR)

But now, looking through the 1e DMG, I can’t find any of this. Ideally, I’d like information on how a Wish spell would interact with ability score increases for each edition of D&D.

What’s the historical interaction between the Wish spell and increasing ability scores?

## How does increasing in size affect adjacent squares and enemies?

Inspired by this question and effectively asking the opposite question. Assuming play is on a grid, what squares can a creature occupy when it goes from being medium size (1×1) to large size (2×2)? Does the square it already occupied when it was medium need to be included in its new form? What are the options for the three additional squares, can they simply be any that would make the creature 2×2? How does increasing in size interact with the “Moving Around Other Creatures” rule which states:

Whether a creature is a friend or an enemy, you can’t willingly end your move in its space.

So if there were a creature in one direction would you not be able to include its current space in your new form/size? What if you were surrounded by creatures, could you increase in size at all?

Some hopefully helpful diagrams: You are C, monsters are X, empty spaces are #.

Can

``###   #C#   ###    ``

change into:

``CC#   CC#   ###   ``

or

``#CC   #CC   ###   ``

What can

``XXX   #C#   ###   ``

change into?

What can happen from this last scenario:

``XXX   XCX   XXX ``

Note: I am looking for an answer that is rooted in RAW, but if no answer exists there an answer from experience with this issue would also work.

Examples of why this might matter:. If you end up occupying the same space as an enemy then a spell like fireball would no longer be able to target you.
If you push the creatures out of the way this could do things such as pushing then into a moonbeam spell.
If you are not allowed to occupy the same space as the monsters then I am confused what would happen if you were initially surrounded.

What squares can you occupy when your size increases, and do other nearby creatures impact this?

## Finding largest farthest increasing pairs in array?

Is there a simple linear algorithm for finding in an array A two such elements that $$A[i] > A[j]$$ and $$i – j$$ is as large as possible? If one wants to actually return the indices, then linear memory usage seems impossible to avoid. If the only result needed is the value $$i – j$$, i.e. the distance, can we get away with less than linear memory?

Obviously, one can use constant memory if one is willing to loop over all pairs (i, j).

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## How does Wild Shape interact with stat increasing items?

I got two sub questions for particular magic items. The first are tomes and manuals which I supposed, are ‘consumed’. Does a druid who just wild shaped can benefit from it (i.e. Dex, Con, Str)?

Second is, for the item like Ioun Stone of Mastery, how would you identify what stat a creature is proficient with to benefit from the item?

Thanks.

## Increasing point size in 3D + color plot

I have a “4D” plot (3D + color) that I want to increase the point size of the points. Seems easy enough right? Just use the following is what I thought:

``PlotStyle -> PointSize[0.01] ``

However, whenever I add this to my code, it makes all the color from my “4th” dimension go away and just turns into one uniform color. How do I get past this?

Also! I’ve had people help me here on Stack Exchange to get to the code I am posting so a lot of credit goes to them for the 3d + color plot. 🙂

Here is my code and a picture of the plot that I want the point sizes to be increased.

``normdSi =    Table[(p - Min[Sidata1[[All, 2]]])/(Max[Sidata1[[All, 2]]] -        Min[Sidata1[[All, 2]]]), {p, Sidata1[[All, 2]]}]; colorsSi = Table[ColorData["BlueGreenYellow"][d], {d, normdSi}]; Si4dplot =   ListPointPlot3D[{#[[3 ;; 5]]} & /@ Sidata1, PlotStyle -> colorsSi,    AxesLabel -> {"q", "\!\(\*SuperscriptBox[\(s\), \(2\)]\)",      "\[Alpha]"}, ImageSize -> Full, LabelStyle -> {18},    ViewPoint -> {-2, -.7, 0}, PlotLabel -> Style["Si", 24]] ``

Here is my data:

``{{0, 0.00586554, 1.85613, 4.76551*10^-28, 5.26926}, {1, 0.00586038,    1.85857, 0.00496208, 5.26553}, {2, 0.00583292, 1.86694, 0.00737286,    5.28264}, {3, 0.00578611, 1.88105, 0.00995626, 5.31438}, {4,    0.00572788, 1.90041, 0.0195179, 5.34518}, {5, 0.0056491, 1.92576,    0.026371, 5.3971}, {6, 0.00555664, 1.95654, 0.0357567, 5.4579}, {7,    0.005454, 1.99256, 0.0488175, 5.52469}, {8, 0.00533414, 2.03443,    0.0587529, 5.61315}, {9, 0.00520924, 2.08121, 0.0728039,    5.70511}, {10, 0.00507624, 2.13333, 0.0868532, 5.80991}, {11,    0.00493048, 2.19114, 0.0969574, 5.937}, {12, 0.00479085, 2.25265,    0.111276, 6.06199}, {13, 0.00464483, 2.31891, 0.12169,    6.20612}, {14, 0.00449348, 2.3895, 0.12767, 6.36945}, {15,    0.00435635, 2.46022, 0.134399, 6.52873}, {16, 0.00421573, 2.53408,    0.13467, 6.70861}, {17, 0.00408066, 2.60798, 0.130038,    6.89764}, {18, 0.003962, 2.67731, 0.122165, 7.08088}, {19,    0.0038446, 2.74659, 0.107359, 7.27855}, {20, 0.0037445, 2.80767,    0.0891022, 7.46372}, {21, 0.00366125, 2.85969, 0.0685708,    7.63264}, {22, 0.00358362, 2.90781, 0.0439095, 7.80201}, {23,    0.00353606, 2.93743, 0.0254055, 7.91445}, {24, 0.00350441, 2.9569,    0.0112019, 7.99332}, {25, 0.00348267, 2.96992, 4.2709*10^-28,    8.05004}, {26, 0.00350619, 2.95655, 0.0140871, 7.98594}, {27,    0.00354397, 2.9359, 0.0380519, 7.88204}, {28, 0.0036054, 2.90368,    0.077481, 7.71557}, {29, 0.00371579, 2.84915, 0.145356,    7.43302}, {30, 0.00384706, 2.78696, 0.22453, 7.10949}, {31,    0.00402225, 2.70695, 0.319264, 6.71886}, {32, 0.00425262, 2.60492,    0.42526, 6.26836}, {33, 0.00451347, 2.4927, 0.533519, 5.80028}, {34,    0.00484984, 2.35888, 0.635259, 5.31261}, {35, 0.0052487, 2.21331,    0.726175, 4.82797}, {36, 0.00569123, 2.06324, 0.807586,    4.35526}, {37, 0.00625698, 1.90488, 0.858761, 3.92207}, {38,    0.00689091, 1.74683, 0.894649, 3.51698}, {39, 0.00759815, 1.59081,    0.915479, 3.14207}, {40, 0.00847396, 1.44051, 0.907596,    2.82513}, {41, 0.00943069, 1.29475, 0.890287, 2.53427}, {42,    0.0105185, 1.15519, 0.860229, 2.27889}, {43, 0.0117991, 1.02388,    0.81632, 2.06547}, {44, 0.0131859, 0.897942, 0.767916,    1.87227}, {45, 0.0147797, 0.780792, 0.713042, 1.70999}, {46,    0.0165753, 0.670983, 0.65441, 1.57258}, {47, 0.0184995, 0.565447,    0.594737, 1.44957}, {48, 0.0207039, 0.467657, 0.533712,    1.35086}, {49, 0.0230828, 0.374668, 0.47279, 1.26641}, {50,    0.025591, 0.285255, 0.412327, 1.1928}, {51, 0.0283735, 0.201725,    0.353629, 1.13931}, {52, 0.0312372, 0.119185, 0.295722,    1.09523}, {53, 0.0341712, 0.0377658, 0.238978, 1.06272}, {54,    0.0370919, -0.0411159, 0.184837, 1.04753}, {55, 0.03991, -0.118284,    0.132156, 1.04359}, {56, 0.0424219, -0.189196, 0.0842067,    1.05721}, {57, 0.0443678, -0.245829, 0.0451427, 1.09405}, {58,    0.0458307, -0.287877, 0.0134487, 1.15053}, {59,    0.0461722, -0.273939, 0.00724523, 1.25192}, {60, 0.04547, -0.189799,    0.029489, 1.40789}, {61, 0.0439536, -0.040825, 0.0760268,    1.61787}, {62, 0.0410509, 0.287135, 0.188407, 1.98092}, {63,    0.0374884, 0.737128, 0.339519, 2.48104}, {64, 0.0334165, 1.32035,    0.529929, 3.15253}, {65, 0.0292259, 2.16715, 0.792678,    4.24807}, {66, 0.0249774, 3.20952, 1.09423, 5.70677}, {67,    0.0209618, 4.54328, 1.4547, 7.76466}, {68, 0.0175535, 6.37734,    1.90226, 10.9599}, {69, 0.0142979, 8.71509, 2.3962, 15.3082}, {70,    0.0116484, 11.9196, 2.96614, 21.8363}, {71, 0.00952234, 16.4455,    3.46263, 31.8198}, {72, 0.00756156, 22.4422, 3.69173, 45.7099}, {73,    0.00637376, 29.2974, 3.10612, 62.8753}, {74, 0.00557852, 35.9338,    1.7877, 80.5131}, {75, 0.0050232, 41.7985, 8.59236*10^-27,    96.93}, {76, 0.00564134, 35.4285, 2.17511, 78.8748}, {77,    0.00664996, 27.396, 4.4568, 56.9122}, {78, 0.00830127, 18.7827,    5.93387, 34.9235}, {79, 0.0112518, 12.1619, 5.34133, 20.6996}, {80,    0.0147477, 7.27342, 4.38667, 11.0649}, {81, 0.0192714, 4.31069,    3.28923, 6.07815}, {82, 0.0249166, 2.64861, 2.36017, 3.78892}, {83,    0.0310646, 1.44104, 1.61259, 2.25868}, {84, 0.038089, 0.763281,    1.11354, 1.51843}, {85, 0.0455155, 0.323334, 0.756884,    1.08172}, {86, 0.0530392, -0.00833097, 0.480004, 0.764547}, {87,    0.0602388, -0.191606, 0.314908, 0.599}, {88, 0.0669456, -0.322307,    0.19112, 0.476188}, {89, 0.0729913, -0.407595, 0.101166,    0.385506}, {90, 0.0773187, -0.422702, 0.0563086, 0.338125}, {91,    0.0808823, -0.415833, 0.0252911, 0.301559}, {92,    0.0833385, -0.387366, 0.00888385, 0.276938}, {93,    0.0845245, -0.347873, 0.00355736, 0.262038}, {94,    0.085323, -0.310296, 0.000716217, 0.249179}, {95,    0.0854916, -0.278922, 0.00021299, 0.239806}, {96,    0.0853257, -0.253975, 0.000358958, 0.232963}, {97,    0.0851545, -0.232974, 0.000267074, 0.227071}, {98,    0.0848481, -0.218693, 0.000354803, 0.223956}, {99,    0.084646, -0.209427, 0.000303089, 0.222006}, {100,    0.0846189, -0.204535, 3.2539*10^-31, 0.220639}, {101,    0.084646, -0.209427, 0.000303089, 0.222006}, {102,    0.0848481, -0.218693, 0.000354803, 0.223956}, {103,    0.0851545, -0.232974, 0.000267074, 0.227071}, {104,    0.0853257, -0.253975, 0.000358958, 0.232963}, {105,    0.0854916, -0.278922, 0.00021299, 0.239806}, {106,    0.085323, -0.310296, 0.000716217, 0.249179}, {107,    0.0845245, -0.347873, 0.00355736, 0.262038}, {108,    0.0833385, -0.387366, 0.00888385, 0.276938}, {109,    0.0808823, -0.415833, 0.0252911, 0.301559}, {110,    0.0773187, -0.422702, 0.0563086, 0.338125}, {111,    0.0729913, -0.407595, 0.101166, 0.385506}, {112,    0.0669456, -0.322307, 0.19112, 0.476188}, {113,    0.0602388, -0.191606, 0.314908, 0.599}, {114,    0.0530392, -0.00833097, 0.480004, 0.764547}, {115, 0.0455155,    0.323334, 0.756884, 1.08172}, {116, 0.038089, 0.763281, 1.11354,    1.51843}, {117, 0.0310646, 1.44104, 1.61259, 2.25868}, {118,    0.0249166, 2.64861, 2.36017, 3.78892}, {119, 0.0192714, 4.31069,    3.28923, 6.07815}, {120, 0.0147477, 7.27342, 4.38667,    11.0649}, {121, 0.0112518, 12.1619, 5.34133, 20.6996}, {122,    0.00830127, 18.7827, 5.93387, 34.9235}, {123, 0.00664996, 27.396,    4.4568, 56.9122}, {124, 0.00564134, 35.4285, 2.17511,    78.8748}, {125, 0.0050232, 41.7985, 5.2584*10^-27, 96.93}, {126,    0.00557852, 35.9338, 1.7877, 80.5131}, {127, 0.00637376, 29.2974,    3.10612, 62.8753}, {128, 0.00756156, 22.4422, 3.69173,    45.7099}, {129, 0.00952234, 16.4455, 3.46263, 31.8198}, {130,    0.0116484, 11.9196, 2.96614, 21.8363}, {131, 0.0142979, 8.71509,    2.3962, 15.3082}, {132, 0.0175535, 6.37734, 1.90226, 10.9599}, {133,    0.0209618, 4.54328, 1.4547, 7.76466}, {134, 0.0249774, 3.20952,    1.09423, 5.70677}, {135, 0.0292259, 2.16715, 0.792678,    4.24807}, {136, 0.0334165, 1.32035, 0.529929, 3.15253}, {137,    0.0374884, 0.737128, 0.339519, 2.48104}, {138, 0.0410509, 0.287135,    0.188407, 1.98092}, {139, 0.0439536, -0.040825, 0.0760268,    1.61787}, {140, 0.04547, -0.189799, 0.029489, 1.40789}, {141,    0.0461722, -0.273939, 0.00724523, 1.25192}, {142,    0.0458307, -0.287877, 0.0134487, 1.15053}, {143,    0.0443678, -0.245829, 0.0451427, 1.09405}, {144,    0.0424219, -0.189196, 0.0842067, 1.05721}, {145, 0.03991, -0.118284,    0.132156, 1.04359}, {146, 0.0370919, -0.0411159, 0.184837,    1.04753}, {147, 0.0341712, 0.0377658, 0.238978, 1.06272}, {148,    0.0312372, 0.119185, 0.295722, 1.09523}, {149, 0.0283735, 0.201725,    0.353629, 1.13931}, {150, 0.025591, 0.285255, 0.412327,    1.1928}, {151, 0.0230828, 0.374668, 0.47279, 1.26641}, {152,    0.0207039, 0.467657, 0.533712, 1.35086}, {153, 0.0184995, 0.565447,    0.594737, 1.44957}, {154, 0.0165753, 0.670983, 0.65441,    1.57258}, {155, 0.0147797, 0.780792, 0.713042, 1.70999}, {156,    0.0131859, 0.897942, 0.767916, 1.87227}, {157, 0.0117991, 1.02388,    0.81632, 2.06547}, {158, 0.0105185, 1.15519, 0.860229,    2.27889}, {159, 0.00943069, 1.29475, 0.890287, 2.53427}, {160,    0.00847396, 1.44051, 0.907596, 2.82513}, {161, 0.00759815, 1.59081,    0.915479, 3.14207}, {162, 0.00689091, 1.74683, 0.894649,    3.51698}, {163, 0.00625698, 1.90488, 0.858761, 3.92207}, {164,    0.00569123, 2.06324, 0.807586, 4.35526}, {165, 0.0052487, 2.21331,    0.726175, 4.82797}, {166, 0.00484984, 2.35888, 0.635259,    5.31261}, {167, 0.00451347, 2.4927, 0.533519, 5.80028}, {168,    0.00425262, 2.60492, 0.42526, 6.26836}, {169, 0.00402225, 2.70695,    0.319264, 6.71886}, {170, 0.00384706, 2.78696, 0.22453,    7.10949}, {171, 0.00371579, 2.84915, 0.145356, 7.43302}, {172,    0.0036054, 2.90368, 0.077481, 7.71557}, {173, 0.00354397, 2.9359,    0.0380519, 7.88204}, {174, 0.00350619, 2.95655, 0.0140871,    7.98594}, {175, 0.00348267, 2.96992, 4.34023*10^-27, 8.05004}, {176,    0.00350441, 2.9569, 0.0112019, 7.99332}, {177, 0.00353606, 2.93743,    0.0254055, 7.91445}, {178, 0.00358362, 2.90781, 0.0439095,    7.80201}, {179, 0.00366125, 2.85969, 0.0685708, 7.63264}, {180,    0.0037445, 2.80767, 0.0891022, 7.46372}, {181, 0.0038446, 2.74659,    0.107359, 7.27855}, {182, 0.003962, 2.67731, 0.122165,    7.08088}, {183, 0.00408066, 2.60798, 0.130038, 6.89764}, {184,    0.00421573, 2.53408, 0.13467, 6.70861}, {185, 0.00435635, 2.46022,    0.134399, 6.52873}, {186, 0.00449348, 2.3895, 0.12767,    6.36945}, {187, 0.00464483, 2.31891, 0.12169, 6.20612}, {188,    0.00479085, 2.25265, 0.111276, 6.06199}, {189, 0.00493048, 2.19114,    0.0969574, 5.937}, {190, 0.00507624, 2.13333, 0.0868532,    5.80991}, {191, 0.00520924, 2.08121, 0.0728039, 5.70511}, {192,    0.00533414, 2.03443, 0.0587529, 5.61315}, {193, 0.005454, 1.99256,    0.0488175, 5.52469}, {194, 0.00555664, 1.95654, 0.0357567,    5.4579}, {195, 0.0056491, 1.92576, 0.026371, 5.3971}, {196,    0.00572788, 1.90041, 0.0195179, 5.34518}, {197, 0.00578611, 1.88105,    0.00995626, 5.31438}, {198, 0.00583292, 1.86694, 0.00737286,    5.28264}, {199, 0.00586038, 1.85857, 0.00496208, 5.26553}, {200,    0.00586554, 1.85613, 4.76551*10^-28, 5.26926}} ``

## How does increasing PCs’ wealth beyond the WBL table affect power disparity between character class tiers?

It’s well-known what happens when characters get less gold than they are supposed toaccording to the wealth-by-level (WBL) guidelines. Basically, the well-known power disparity between casters and non-casters becomes even stronger: money is Fighter’s access to magic, and magic is true power in Pathfinder. Without magic, the Fighter has significantly less power.

However, what are the consequences of the party getting significantly more money than it’s supposed to? E.g. doubling WBL, so a level 4 characters would get 12.000 gp worth of valuables instead of only 6.000.

Of course, this will make the affected characters more powerful, and they will require harder challenges to have meaningful encounters. But will the power disparity be affected in any way?

## Is it possible to calculate the longest increasing subsequence of an unsorted list

The problem goes like this:

Instance: List with indices and values

Question: Is it possible to calculate the LIS by always taking the next element from the left (without sorting the indices of the instance)?

Example:

Index array:…………1 2 3 4 5 6

Indices instance:….1 3 5 2 4 6

Values instance:…..2 4 6 1 3 5

The longest increasing subsequences would now be:

245 246 235 236

145 146 135 136

## Why increasing the beam width does not necessarily improve the beam search solution?

Normally increasing the beam width is a better solution, but I have read that this is not necessarily true, is there any proof of this?